src/HOL/Divides.thy
 author webertj Tue Nov 17 10:17:53 2009 +0000 (2009-11-17) changeset 33728 cb4235333c30 parent 33364 2bd12592c5e8 child 33730 1755ca4ec022 permissions -rw-r--r--
Fixed splitting of div and mod on integers (split theorem differed from implementation).
     1 (*  Title:      HOL/Divides.thy

     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

     3     Copyright   1999  University of Cambridge

     4 *)

     5

     6 header {* The division operators div and mod *}

     7

     8 theory Divides

     9 imports Nat_Numeral Nat_Transfer

    10 uses "~~/src/Provers/Arith/cancel_div_mod.ML"

    11 begin

    12

    13 subsection {* Syntactic division operations *}

    14

    15 class div = dvd +

    16   fixes div :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "div" 70)

    17     and mod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "mod" 70)

    18

    19

    20 subsection {* Abstract division in commutative semirings. *}

    21

    22 class semiring_div = comm_semiring_1_cancel + no_zero_divisors + div +

    23   assumes mod_div_equality: "a div b * b + a mod b = a"

    24     and div_by_0 [simp]: "a div 0 = 0"

    25     and div_0 [simp]: "0 div a = 0"

    26     and div_mult_self1 [simp]: "b \<noteq> 0 \<Longrightarrow> (a + c * b) div b = c + a div b"

    27     and div_mult_mult1 [simp]: "c \<noteq> 0 \<Longrightarrow> (c * a) div (c * b) = a div b"

    28 begin

    29

    30 text {* @{const div} and @{const mod} *}

    31

    32 lemma mod_div_equality2: "b * (a div b) + a mod b = a"

    33   unfolding mult_commute [of b]

    34   by (rule mod_div_equality)

    35

    36 lemma mod_div_equality': "a mod b + a div b * b = a"

    37   using mod_div_equality [of a b]

    38   by (simp only: add_ac)

    39

    40 lemma div_mod_equality: "((a div b) * b + a mod b) + c = a + c"

    41   by (simp add: mod_div_equality)

    42

    43 lemma div_mod_equality2: "(b * (a div b) + a mod b) + c = a + c"

    44   by (simp add: mod_div_equality2)

    45

    46 lemma mod_by_0 [simp]: "a mod 0 = a"

    47   using mod_div_equality [of a zero] by simp

    48

    49 lemma mod_0 [simp]: "0 mod a = 0"

    50   using mod_div_equality [of zero a] div_0 by simp

    51

    52 lemma div_mult_self2 [simp]:

    53   assumes "b \<noteq> 0"

    54   shows "(a + b * c) div b = c + a div b"

    55   using assms div_mult_self1 [of b a c] by (simp add: mult_commute)

    56

    57 lemma mod_mult_self1 [simp]: "(a + c * b) mod b = a mod b"

    58 proof (cases "b = 0")

    59   case True then show ?thesis by simp

    60 next

    61   case False

    62   have "a + c * b = (a + c * b) div b * b + (a + c * b) mod b"

    63     by (simp add: mod_div_equality)

    64   also from False div_mult_self1 [of b a c] have

    65     "\<dots> = (c + a div b) * b + (a + c * b) mod b"

    66       by (simp add: algebra_simps)

    67   finally have "a = a div b * b + (a + c * b) mod b"

    68     by (simp add: add_commute [of a] add_assoc left_distrib)

    69   then have "a div b * b + (a + c * b) mod b = a div b * b + a mod b"

    70     by (simp add: mod_div_equality)

    71   then show ?thesis by simp

    72 qed

    73

    74 lemma mod_mult_self2 [simp]: "(a + b * c) mod b = a mod b"

    75   by (simp add: mult_commute [of b])

    76

    77 lemma div_mult_self1_is_id [simp]: "b \<noteq> 0 \<Longrightarrow> b * a div b = a"

    78   using div_mult_self2 [of b 0 a] by simp

    79

    80 lemma div_mult_self2_is_id [simp]: "b \<noteq> 0 \<Longrightarrow> a * b div b = a"

    81   using div_mult_self1 [of b 0 a] by simp

    82

    83 lemma mod_mult_self1_is_0 [simp]: "b * a mod b = 0"

    84   using mod_mult_self2 [of 0 b a] by simp

    85

    86 lemma mod_mult_self2_is_0 [simp]: "a * b mod b = 0"

    87   using mod_mult_self1 [of 0 a b] by simp

    88

    89 lemma div_by_1 [simp]: "a div 1 = a"

    90   using div_mult_self2_is_id [of 1 a] zero_neq_one by simp

    91

    92 lemma mod_by_1 [simp]: "a mod 1 = 0"

    93 proof -

    94   from mod_div_equality [of a one] div_by_1 have "a + a mod 1 = a" by simp

    95   then have "a + a mod 1 = a + 0" by simp

    96   then show ?thesis by (rule add_left_imp_eq)

    97 qed

    98

    99 lemma mod_self [simp]: "a mod a = 0"

   100   using mod_mult_self2_is_0 [of 1] by simp

   101

   102 lemma div_self [simp]: "a \<noteq> 0 \<Longrightarrow> a div a = 1"

   103   using div_mult_self2_is_id [of _ 1] by simp

   104

   105 lemma div_add_self1 [simp]:

   106   assumes "b \<noteq> 0"

   107   shows "(b + a) div b = a div b + 1"

   108   using assms div_mult_self1 [of b a 1] by (simp add: add_commute)

   109

   110 lemma div_add_self2 [simp]:

   111   assumes "b \<noteq> 0"

   112   shows "(a + b) div b = a div b + 1"

   113   using assms div_add_self1 [of b a] by (simp add: add_commute)

   114

   115 lemma mod_add_self1 [simp]:

   116   "(b + a) mod b = a mod b"

   117   using mod_mult_self1 [of a 1 b] by (simp add: add_commute)

   118

   119 lemma mod_add_self2 [simp]:

   120   "(a + b) mod b = a mod b"

   121   using mod_mult_self1 [of a 1 b] by simp

   122

   123 lemma mod_div_decomp:

   124   fixes a b

   125   obtains q r where "q = a div b" and "r = a mod b"

   126     and "a = q * b + r"

   127 proof -

   128   from mod_div_equality have "a = a div b * b + a mod b" by simp

   129   moreover have "a div b = a div b" ..

   130   moreover have "a mod b = a mod b" ..

   131   note that ultimately show thesis by blast

   132 qed

   133

   134 lemma dvd_eq_mod_eq_0 [code, code_unfold, code_inline del]: "a dvd b \<longleftrightarrow> b mod a = 0"

   135 proof

   136   assume "b mod a = 0"

   137   with mod_div_equality [of b a] have "b div a * a = b" by simp

   138   then have "b = a * (b div a)" unfolding mult_commute ..

   139   then have "\<exists>c. b = a * c" ..

   140   then show "a dvd b" unfolding dvd_def .

   141 next

   142   assume "a dvd b"

   143   then have "\<exists>c. b = a * c" unfolding dvd_def .

   144   then obtain c where "b = a * c" ..

   145   then have "b mod a = a * c mod a" by simp

   146   then have "b mod a = c * a mod a" by (simp add: mult_commute)

   147   then show "b mod a = 0" by simp

   148 qed

   149

   150 lemma mod_div_trivial [simp]: "a mod b div b = 0"

   151 proof (cases "b = 0")

   152   assume "b = 0"

   153   thus ?thesis by simp

   154 next

   155   assume "b \<noteq> 0"

   156   hence "a div b + a mod b div b = (a mod b + a div b * b) div b"

   157     by (rule div_mult_self1 [symmetric])

   158   also have "\<dots> = a div b"

   159     by (simp only: mod_div_equality')

   160   also have "\<dots> = a div b + 0"

   161     by simp

   162   finally show ?thesis

   163     by (rule add_left_imp_eq)

   164 qed

   165

   166 lemma mod_mod_trivial [simp]: "a mod b mod b = a mod b"

   167 proof -

   168   have "a mod b mod b = (a mod b + a div b * b) mod b"

   169     by (simp only: mod_mult_self1)

   170   also have "\<dots> = a mod b"

   171     by (simp only: mod_div_equality')

   172   finally show ?thesis .

   173 qed

   174

   175 lemma dvd_imp_mod_0: "a dvd b \<Longrightarrow> b mod a = 0"

   176 by (rule dvd_eq_mod_eq_0[THEN iffD1])

   177

   178 lemma dvd_div_mult_self: "a dvd b \<Longrightarrow> (b div a) * a = b"

   179 by (subst (2) mod_div_equality [of b a, symmetric]) (simp add:dvd_imp_mod_0)

   180

   181 lemma dvd_mult_div_cancel: "a dvd b \<Longrightarrow> a * (b div a) = b"

   182 by (drule dvd_div_mult_self) (simp add: mult_commute)

   183

   184 lemma dvd_div_mult: "a dvd b \<Longrightarrow> (b div a) * c = b * c div a"

   185 apply (cases "a = 0")

   186  apply simp

   187 apply (auto simp: dvd_def mult_assoc)

   188 done

   189

   190 lemma div_dvd_div[simp]:

   191   "a dvd b \<Longrightarrow> a dvd c \<Longrightarrow> (b div a dvd c div a) = (b dvd c)"

   192 apply (cases "a = 0")

   193  apply simp

   194 apply (unfold dvd_def)

   195 apply auto

   196  apply(blast intro:mult_assoc[symmetric])

   197 apply(fastsimp simp add: mult_assoc)

   198 done

   199

   200 lemma dvd_mod_imp_dvd: "[| k dvd m mod n;  k dvd n |] ==> k dvd m"

   201   apply (subgoal_tac "k dvd (m div n) *n + m mod n")

   202    apply (simp add: mod_div_equality)

   203   apply (simp only: dvd_add dvd_mult)

   204   done

   205

   206 text {* Addition respects modular equivalence. *}

   207

   208 lemma mod_add_left_eq: "(a + b) mod c = (a mod c + b) mod c"

   209 proof -

   210   have "(a + b) mod c = (a div c * c + a mod c + b) mod c"

   211     by (simp only: mod_div_equality)

   212   also have "\<dots> = (a mod c + b + a div c * c) mod c"

   213     by (simp only: add_ac)

   214   also have "\<dots> = (a mod c + b) mod c"

   215     by (rule mod_mult_self1)

   216   finally show ?thesis .

   217 qed

   218

   219 lemma mod_add_right_eq: "(a + b) mod c = (a + b mod c) mod c"

   220 proof -

   221   have "(a + b) mod c = (a + (b div c * c + b mod c)) mod c"

   222     by (simp only: mod_div_equality)

   223   also have "\<dots> = (a + b mod c + b div c * c) mod c"

   224     by (simp only: add_ac)

   225   also have "\<dots> = (a + b mod c) mod c"

   226     by (rule mod_mult_self1)

   227   finally show ?thesis .

   228 qed

   229

   230 lemma mod_add_eq: "(a + b) mod c = (a mod c + b mod c) mod c"

   231 by (rule trans [OF mod_add_left_eq mod_add_right_eq])

   232

   233 lemma mod_add_cong:

   234   assumes "a mod c = a' mod c"

   235   assumes "b mod c = b' mod c"

   236   shows "(a + b) mod c = (a' + b') mod c"

   237 proof -

   238   have "(a mod c + b mod c) mod c = (a' mod c + b' mod c) mod c"

   239     unfolding assms ..

   240   thus ?thesis

   241     by (simp only: mod_add_eq [symmetric])

   242 qed

   243

   244 lemma div_add [simp]: "z dvd x \<Longrightarrow> z dvd y

   245   \<Longrightarrow> (x + y) div z = x div z + y div z"

   246 by (cases "z = 0", simp, unfold dvd_def, auto simp add: algebra_simps)

   247

   248 text {* Multiplication respects modular equivalence. *}

   249

   250 lemma mod_mult_left_eq: "(a * b) mod c = ((a mod c) * b) mod c"

   251 proof -

   252   have "(a * b) mod c = ((a div c * c + a mod c) * b) mod c"

   253     by (simp only: mod_div_equality)

   254   also have "\<dots> = (a mod c * b + a div c * b * c) mod c"

   255     by (simp only: algebra_simps)

   256   also have "\<dots> = (a mod c * b) mod c"

   257     by (rule mod_mult_self1)

   258   finally show ?thesis .

   259 qed

   260

   261 lemma mod_mult_right_eq: "(a * b) mod c = (a * (b mod c)) mod c"

   262 proof -

   263   have "(a * b) mod c = (a * (b div c * c + b mod c)) mod c"

   264     by (simp only: mod_div_equality)

   265   also have "\<dots> = (a * (b mod c) + a * (b div c) * c) mod c"

   266     by (simp only: algebra_simps)

   267   also have "\<dots> = (a * (b mod c)) mod c"

   268     by (rule mod_mult_self1)

   269   finally show ?thesis .

   270 qed

   271

   272 lemma mod_mult_eq: "(a * b) mod c = ((a mod c) * (b mod c)) mod c"

   273 by (rule trans [OF mod_mult_left_eq mod_mult_right_eq])

   274

   275 lemma mod_mult_cong:

   276   assumes "a mod c = a' mod c"

   277   assumes "b mod c = b' mod c"

   278   shows "(a * b) mod c = (a' * b') mod c"

   279 proof -

   280   have "(a mod c * (b mod c)) mod c = (a' mod c * (b' mod c)) mod c"

   281     unfolding assms ..

   282   thus ?thesis

   283     by (simp only: mod_mult_eq [symmetric])

   284 qed

   285

   286 lemma mod_mod_cancel:

   287   assumes "c dvd b"

   288   shows "a mod b mod c = a mod c"

   289 proof -

   290   from c dvd b obtain k where "b = c * k"

   291     by (rule dvdE)

   292   have "a mod b mod c = a mod (c * k) mod c"

   293     by (simp only: b = c * k)

   294   also have "\<dots> = (a mod (c * k) + a div (c * k) * k * c) mod c"

   295     by (simp only: mod_mult_self1)

   296   also have "\<dots> = (a div (c * k) * (c * k) + a mod (c * k)) mod c"

   297     by (simp only: add_ac mult_ac)

   298   also have "\<dots> = a mod c"

   299     by (simp only: mod_div_equality)

   300   finally show ?thesis .

   301 qed

   302

   303 lemma div_mult_div_if_dvd:

   304   "y dvd x \<Longrightarrow> z dvd w \<Longrightarrow> (x div y) * (w div z) = (x * w) div (y * z)"

   305   apply (cases "y = 0", simp)

   306   apply (cases "z = 0", simp)

   307   apply (auto elim!: dvdE simp add: algebra_simps)

   308   apply (subst mult_assoc [symmetric])

   309   apply (simp add: no_zero_divisors)

   310   done

   311

   312 lemma div_mult_mult2 [simp]:

   313   "c \<noteq> 0 \<Longrightarrow> (a * c) div (b * c) = a div b"

   314   by (drule div_mult_mult1) (simp add: mult_commute)

   315

   316 lemma div_mult_mult1_if [simp]:

   317   "(c * a) div (c * b) = (if c = 0 then 0 else a div b)"

   318   by simp_all

   319

   320 lemma mod_mult_mult1:

   321   "(c * a) mod (c * b) = c * (a mod b)"

   322 proof (cases "c = 0")

   323   case True then show ?thesis by simp

   324 next

   325   case False

   326   from mod_div_equality

   327   have "((c * a) div (c * b)) * (c * b) + (c * a) mod (c * b) = c * a" .

   328   with False have "c * ((a div b) * b + a mod b) + (c * a) mod (c * b)

   329     = c * a + c * (a mod b)" by (simp add: algebra_simps)

   330   with mod_div_equality show ?thesis by simp

   331 qed

   332

   333 lemma mod_mult_mult2:

   334   "(a * c) mod (b * c) = (a mod b) * c"

   335   using mod_mult_mult1 [of c a b] by (simp add: mult_commute)

   336

   337 lemma dvd_mod: "k dvd m \<Longrightarrow> k dvd n \<Longrightarrow> k dvd (m mod n)"

   338   unfolding dvd_def by (auto simp add: mod_mult_mult1)

   339

   340 lemma dvd_mod_iff: "k dvd n \<Longrightarrow> k dvd (m mod n) \<longleftrightarrow> k dvd m"

   341 by (blast intro: dvd_mod_imp_dvd dvd_mod)

   342

   343 lemma div_power:

   344   "y dvd x \<Longrightarrow> (x div y) ^ n = x ^ n div y ^ n"

   345 apply (induct n)

   346  apply simp

   347 apply(simp add: div_mult_div_if_dvd dvd_power_same)

   348 done

   349

   350 end

   351

   352 class ring_div = semiring_div + idom

   353 begin

   354

   355 text {* Negation respects modular equivalence. *}

   356

   357 lemma mod_minus_eq: "(- a) mod b = (- (a mod b)) mod b"

   358 proof -

   359   have "(- a) mod b = (- (a div b * b + a mod b)) mod b"

   360     by (simp only: mod_div_equality)

   361   also have "\<dots> = (- (a mod b) + - (a div b) * b) mod b"

   362     by (simp only: minus_add_distrib minus_mult_left add_ac)

   363   also have "\<dots> = (- (a mod b)) mod b"

   364     by (rule mod_mult_self1)

   365   finally show ?thesis .

   366 qed

   367

   368 lemma mod_minus_cong:

   369   assumes "a mod b = a' mod b"

   370   shows "(- a) mod b = (- a') mod b"

   371 proof -

   372   have "(- (a mod b)) mod b = (- (a' mod b)) mod b"

   373     unfolding assms ..

   374   thus ?thesis

   375     by (simp only: mod_minus_eq [symmetric])

   376 qed

   377

   378 text {* Subtraction respects modular equivalence. *}

   379

   380 lemma mod_diff_left_eq: "(a - b) mod c = (a mod c - b) mod c"

   381   unfolding diff_minus

   382   by (intro mod_add_cong mod_minus_cong) simp_all

   383

   384 lemma mod_diff_right_eq: "(a - b) mod c = (a - b mod c) mod c"

   385   unfolding diff_minus

   386   by (intro mod_add_cong mod_minus_cong) simp_all

   387

   388 lemma mod_diff_eq: "(a - b) mod c = (a mod c - b mod c) mod c"

   389   unfolding diff_minus

   390   by (intro mod_add_cong mod_minus_cong) simp_all

   391

   392 lemma mod_diff_cong:

   393   assumes "a mod c = a' mod c"

   394   assumes "b mod c = b' mod c"

   395   shows "(a - b) mod c = (a' - b') mod c"

   396   unfolding diff_minus using assms

   397   by (intro mod_add_cong mod_minus_cong)

   398

   399 lemma dvd_neg_div: "y dvd x \<Longrightarrow> -x div y = - (x div y)"

   400 apply (case_tac "y = 0") apply simp

   401 apply (auto simp add: dvd_def)

   402 apply (subgoal_tac "-(y * k) = y * - k")

   403  apply (erule ssubst)

   404  apply (erule div_mult_self1_is_id)

   405 apply simp

   406 done

   407

   408 lemma dvd_div_neg: "y dvd x \<Longrightarrow> x div -y = - (x div y)"

   409 apply (case_tac "y = 0") apply simp

   410 apply (auto simp add: dvd_def)

   411 apply (subgoal_tac "y * k = -y * -k")

   412  apply (erule ssubst)

   413  apply (rule div_mult_self1_is_id)

   414  apply simp

   415 apply simp

   416 done

   417

   418 end

   419

   420

   421 subsection {* Division on @{typ nat} *}

   422

   423 text {*

   424   We define @{const div} and @{const mod} on @{typ nat} by means

   425   of a characteristic relation with two input arguments

   426   @{term "m\<Colon>nat"}, @{term "n\<Colon>nat"} and two output arguments

   427   @{term "q\<Colon>nat"}(uotient) and @{term "r\<Colon>nat"}(emainder).

   428 *}

   429

   430 definition divmod_nat_rel :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat \<Rightarrow> bool" where

   431   "divmod_nat_rel m n qr \<longleftrightarrow>

   432     m = fst qr * n + snd qr \<and>

   433       (if n = 0 then fst qr = 0 else if n > 0 then 0 \<le> snd qr \<and> snd qr < n else n < snd qr \<and> snd qr \<le> 0)"

   434

   435 text {* @{const divmod_nat_rel} is total: *}

   436

   437 lemma divmod_nat_rel_ex:

   438   obtains q r where "divmod_nat_rel m n (q, r)"

   439 proof (cases "n = 0")

   440   case True  with that show thesis

   441     by (auto simp add: divmod_nat_rel_def)

   442 next

   443   case False

   444   have "\<exists>q r. m = q * n + r \<and> r < n"

   445   proof (induct m)

   446     case 0 with n \<noteq> 0

   447     have "(0\<Colon>nat) = 0 * n + 0 \<and> 0 < n" by simp

   448     then show ?case by blast

   449   next

   450     case (Suc m) then obtain q' r'

   451       where m: "m = q' * n + r'" and n: "r' < n" by auto

   452     then show ?case proof (cases "Suc r' < n")

   453       case True

   454       from m n have "Suc m = q' * n + Suc r'" by simp

   455       with True show ?thesis by blast

   456     next

   457       case False then have "n \<le> Suc r'" by auto

   458       moreover from n have "Suc r' \<le> n" by auto

   459       ultimately have "n = Suc r'" by auto

   460       with m have "Suc m = Suc q' * n + 0" by simp

   461       with n \<noteq> 0 show ?thesis by blast

   462     qed

   463   qed

   464   with that show thesis

   465     using n \<noteq> 0 by (auto simp add: divmod_nat_rel_def)

   466 qed

   467

   468 text {* @{const divmod_nat_rel} is injective: *}

   469

   470 lemma divmod_nat_rel_unique:

   471   assumes "divmod_nat_rel m n qr"

   472     and "divmod_nat_rel m n qr'"

   473   shows "qr = qr'"

   474 proof (cases "n = 0")

   475   case True with assms show ?thesis

   476     by (cases qr, cases qr')

   477       (simp add: divmod_nat_rel_def)

   478 next

   479   case False

   480   have aux: "\<And>q r q' r'. q' * n + r' = q * n + r \<Longrightarrow> r < n \<Longrightarrow> q' \<le> (q\<Colon>nat)"

   481   apply (rule leI)

   482   apply (subst less_iff_Suc_add)

   483   apply (auto simp add: add_mult_distrib)

   484   done

   485   from n \<noteq> 0 assms have "fst qr = fst qr'"

   486     by (auto simp add: divmod_nat_rel_def intro: order_antisym dest: aux sym)

   487   moreover from this assms have "snd qr = snd qr'"

   488     by (simp add: divmod_nat_rel_def)

   489   ultimately show ?thesis by (cases qr, cases qr') simp

   490 qed

   491

   492 text {*

   493   We instantiate divisibility on the natural numbers by

   494   means of @{const divmod_nat_rel}:

   495 *}

   496

   497 instantiation nat :: semiring_div

   498 begin

   499

   500 definition divmod_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat" where

   501   [code del]: "divmod_nat m n = (THE qr. divmod_nat_rel m n qr)"

   502

   503 lemma divmod_nat_rel_divmod_nat:

   504   "divmod_nat_rel m n (divmod_nat m n)"

   505 proof -

   506   from divmod_nat_rel_ex

   507     obtain qr where rel: "divmod_nat_rel m n qr" .

   508   then show ?thesis

   509   by (auto simp add: divmod_nat_def intro: theI elim: divmod_nat_rel_unique)

   510 qed

   511

   512 lemma divmod_nat_eq:

   513   assumes "divmod_nat_rel m n qr"

   514   shows "divmod_nat m n = qr"

   515   using assms by (auto intro: divmod_nat_rel_unique divmod_nat_rel_divmod_nat)

   516

   517 definition div_nat where

   518   "m div n = fst (divmod_nat m n)"

   519

   520 definition mod_nat where

   521   "m mod n = snd (divmod_nat m n)"

   522

   523 lemma divmod_nat_div_mod:

   524   "divmod_nat m n = (m div n, m mod n)"

   525   unfolding div_nat_def mod_nat_def by simp

   526

   527 lemma div_eq:

   528   assumes "divmod_nat_rel m n (q, r)"

   529   shows "m div n = q"

   530   using assms by (auto dest: divmod_nat_eq simp add: divmod_nat_div_mod)

   531

   532 lemma mod_eq:

   533   assumes "divmod_nat_rel m n (q, r)"

   534   shows "m mod n = r"

   535   using assms by (auto dest: divmod_nat_eq simp add: divmod_nat_div_mod)

   536

   537 lemma divmod_nat_rel: "divmod_nat_rel m n (m div n, m mod n)"

   538   by (simp add: div_nat_def mod_nat_def divmod_nat_rel_divmod_nat)

   539

   540 lemma divmod_nat_zero:

   541   "divmod_nat m 0 = (0, m)"

   542 proof -

   543   from divmod_nat_rel [of m 0] show ?thesis

   544     unfolding divmod_nat_div_mod divmod_nat_rel_def by simp

   545 qed

   546

   547 lemma divmod_nat_base:

   548   assumes "m < n"

   549   shows "divmod_nat m n = (0, m)"

   550 proof -

   551   from divmod_nat_rel [of m n] show ?thesis

   552     unfolding divmod_nat_div_mod divmod_nat_rel_def

   553     using assms by (cases "m div n = 0")

   554       (auto simp add: gr0_conv_Suc [of "m div n"])

   555 qed

   556

   557 lemma divmod_nat_step:

   558   assumes "0 < n" and "n \<le> m"

   559   shows "divmod_nat m n = (Suc ((m - n) div n), (m - n) mod n)"

   560 proof -

   561   from divmod_nat_rel have divmod_nat_m_n: "divmod_nat_rel m n (m div n, m mod n)" .

   562   with assms have m_div_n: "m div n \<ge> 1"

   563     by (cases "m div n") (auto simp add: divmod_nat_rel_def)

   564   from assms divmod_nat_m_n have "divmod_nat_rel (m - n) n (m div n - Suc 0, m mod n)"

   565     by (cases "m div n") (auto simp add: divmod_nat_rel_def)

   566   with divmod_nat_eq have "divmod_nat (m - n) n = (m div n - Suc 0, m mod n)" by simp

   567   moreover from divmod_nat_div_mod have "divmod_nat (m - n) n = ((m - n) div n, (m - n) mod n)" .

   568   ultimately have "m div n = Suc ((m - n) div n)"

   569     and "m mod n = (m - n) mod n" using m_div_n by simp_all

   570   then show ?thesis using divmod_nat_div_mod by simp

   571 qed

   572

   573 text {* The ''recursion'' equations for @{const div} and @{const mod} *}

   574

   575 lemma div_less [simp]:

   576   fixes m n :: nat

   577   assumes "m < n"

   578   shows "m div n = 0"

   579   using assms divmod_nat_base divmod_nat_div_mod by simp

   580

   581 lemma le_div_geq:

   582   fixes m n :: nat

   583   assumes "0 < n" and "n \<le> m"

   584   shows "m div n = Suc ((m - n) div n)"

   585   using assms divmod_nat_step divmod_nat_div_mod by simp

   586

   587 lemma mod_less [simp]:

   588   fixes m n :: nat

   589   assumes "m < n"

   590   shows "m mod n = m"

   591   using assms divmod_nat_base divmod_nat_div_mod by simp

   592

   593 lemma le_mod_geq:

   594   fixes m n :: nat

   595   assumes "n \<le> m"

   596   shows "m mod n = (m - n) mod n"

   597   using assms divmod_nat_step divmod_nat_div_mod by (cases "n = 0") simp_all

   598

   599 instance proof -

   600   have [simp]: "\<And>n::nat. n div 0 = 0"

   601     by (simp add: div_nat_def divmod_nat_zero)

   602   have [simp]: "\<And>n::nat. 0 div n = 0"

   603   proof -

   604     fix n :: nat

   605     show "0 div n = 0"

   606       by (cases "n = 0") simp_all

   607   qed

   608   show "OFCLASS(nat, semiring_div_class)" proof

   609     fix m n :: nat

   610     show "m div n * n + m mod n = m"

   611       using divmod_nat_rel [of m n] by (simp add: divmod_nat_rel_def)

   612   next

   613     fix m n q :: nat

   614     assume "n \<noteq> 0"

   615     then show "(q + m * n) div n = m + q div n"

   616       by (induct m) (simp_all add: le_div_geq)

   617   next

   618     fix m n q :: nat

   619     assume "m \<noteq> 0"

   620     then show "(m * n) div (m * q) = n div q"

   621     proof (cases "n \<noteq> 0 \<and> q \<noteq> 0")

   622       case False then show ?thesis by auto

   623     next

   624       case True with m \<noteq> 0

   625         have "m > 0" and "n > 0" and "q > 0" by auto

   626       then have "\<And>a b. divmod_nat_rel n q (a, b) \<Longrightarrow> divmod_nat_rel (m * n) (m * q) (a, m * b)"

   627         by (auto simp add: divmod_nat_rel_def) (simp_all add: algebra_simps)

   628       moreover from divmod_nat_rel have "divmod_nat_rel n q (n div q, n mod q)" .

   629       ultimately have "divmod_nat_rel (m * n) (m * q) (n div q, m * (n mod q))" .

   630       then show ?thesis by (simp add: div_eq)

   631     qed

   632   qed simp_all

   633 qed

   634

   635 end

   636

   637 lemma divmod_nat_if [code]: "divmod_nat m n = (if n = 0 \<or> m < n then (0, m) else

   638   let (q, r) = divmod_nat (m - n) n in (Suc q, r))"

   639 by (simp add: divmod_nat_zero divmod_nat_base divmod_nat_step)

   640     (simp add: divmod_nat_div_mod)

   641

   642 text {* Simproc for cancelling @{const div} and @{const mod} *}

   643

   644 ML {*

   645 local

   646

   647 structure CancelDivMod = CancelDivModFun(struct

   648

   649   val div_name = @{const_name div};

   650   val mod_name = @{const_name mod};

   651   val mk_binop = HOLogic.mk_binop;

   652   val mk_sum = Nat_Arith.mk_sum;

   653   val dest_sum = Nat_Arith.dest_sum;

   654

   655   val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}];

   656

   657   val trans = trans;

   658

   659   val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac

   660     (@{thm monoid_add_class.add_0_left} :: @{thm monoid_add_class.add_0_right} :: @{thms add_ac}))

   661

   662 end)

   663

   664 in

   665

   666 val cancel_div_mod_nat_proc = Simplifier.simproc @{theory}

   667   "cancel_div_mod" ["(m::nat) + n"] (K CancelDivMod.proc);

   668

   669 val _ = Addsimprocs [cancel_div_mod_nat_proc];

   670

   671 end

   672 *}

   673

   674

   675 subsubsection {* Quotient *}

   676

   677 lemma div_geq: "0 < n \<Longrightarrow>  \<not> m < n \<Longrightarrow> m div n = Suc ((m - n) div n)"

   678 by (simp add: le_div_geq linorder_not_less)

   679

   680 lemma div_if: "0 < n \<Longrightarrow> m div n = (if m < n then 0 else Suc ((m - n) div n))"

   681 by (simp add: div_geq)

   682

   683 lemma div_mult_self_is_m [simp]: "0<n ==> (m*n) div n = (m::nat)"

   684 by simp

   685

   686 lemma div_mult_self1_is_m [simp]: "0<n ==> (n*m) div n = (m::nat)"

   687 by simp

   688

   689

   690 subsubsection {* Remainder *}

   691

   692 lemma mod_less_divisor [simp]:

   693   fixes m n :: nat

   694   assumes "n > 0"

   695   shows "m mod n < (n::nat)"

   696   using assms divmod_nat_rel [of m n] unfolding divmod_nat_rel_def by auto

   697

   698 lemma mod_less_eq_dividend [simp]:

   699   fixes m n :: nat

   700   shows "m mod n \<le> m"

   701 proof (rule add_leD2)

   702   from mod_div_equality have "m div n * n + m mod n = m" .

   703   then show "m div n * n + m mod n \<le> m" by auto

   704 qed

   705

   706 lemma mod_geq: "\<not> m < (n\<Colon>nat) \<Longrightarrow> m mod n = (m - n) mod n"

   707 by (simp add: le_mod_geq linorder_not_less)

   708

   709 lemma mod_if: "m mod (n\<Colon>nat) = (if m < n then m else (m - n) mod n)"

   710 by (simp add: le_mod_geq)

   711

   712 lemma mod_1 [simp]: "m mod Suc 0 = 0"

   713 by (induct m) (simp_all add: mod_geq)

   714

   715 lemma mod_mult_distrib: "(m mod n) * (k\<Colon>nat) = (m * k) mod (n * k)"

   716   apply (cases "n = 0", simp)

   717   apply (cases "k = 0", simp)

   718   apply (induct m rule: nat_less_induct)

   719   apply (subst mod_if, simp)

   720   apply (simp add: mod_geq diff_mult_distrib)

   721   done

   722

   723 lemma mod_mult_distrib2: "(k::nat) * (m mod n) = (k*m) mod (k*n)"

   724 by (simp add: mult_commute [of k] mod_mult_distrib)

   725

   726 (* a simple rearrangement of mod_div_equality: *)

   727 lemma mult_div_cancel: "(n::nat) * (m div n) = m - (m mod n)"

   728 by (cut_tac a = m and b = n in mod_div_equality2, arith)

   729

   730 lemma mod_le_divisor[simp]: "0 < n \<Longrightarrow> m mod n \<le> (n::nat)"

   731   apply (drule mod_less_divisor [where m = m])

   732   apply simp

   733   done

   734

   735 subsubsection {* Quotient and Remainder *}

   736

   737 lemma divmod_nat_rel_mult1_eq:

   738   "divmod_nat_rel b c (q, r) \<Longrightarrow> c > 0

   739    \<Longrightarrow> divmod_nat_rel (a * b) c (a * q + a * r div c, a * r mod c)"

   740 by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)

   741

   742 lemma div_mult1_eq:

   743   "(a * b) div c = a * (b div c) + a * (b mod c) div (c::nat)"

   744 apply (cases "c = 0", simp)

   745 apply (blast intro: divmod_nat_rel [THEN divmod_nat_rel_mult1_eq, THEN div_eq])

   746 done

   747

   748 lemma divmod_nat_rel_add1_eq:

   749   "divmod_nat_rel a c (aq, ar) \<Longrightarrow> divmod_nat_rel b c (bq, br) \<Longrightarrow>  c > 0

   750    \<Longrightarrow> divmod_nat_rel (a + b) c (aq + bq + (ar + br) div c, (ar + br) mod c)"

   751 by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)

   752

   753 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)

   754 lemma div_add1_eq:

   755   "(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)"

   756 apply (cases "c = 0", simp)

   757 apply (blast intro: divmod_nat_rel_add1_eq [THEN div_eq] divmod_nat_rel)

   758 done

   759

   760 lemma mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c"

   761   apply (cut_tac m = q and n = c in mod_less_divisor)

   762   apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)

   763   apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst)

   764   apply (simp add: add_mult_distrib2)

   765   done

   766

   767 lemma divmod_nat_rel_mult2_eq:

   768   "divmod_nat_rel a b (q, r) \<Longrightarrow> 0 < b \<Longrightarrow> 0 < c

   769    \<Longrightarrow> divmod_nat_rel a (b * c) (q div c, b *(q mod c) + r)"

   770 by (auto simp add: mult_ac divmod_nat_rel_def add_mult_distrib2 [symmetric] mod_lemma)

   771

   772 lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)"

   773   apply (cases "b = 0", simp)

   774   apply (cases "c = 0", simp)

   775   apply (force simp add: divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN div_eq])

   776   done

   777

   778 lemma mod_mult2_eq: "a mod (b*c) = b*(a div b mod c) + a mod (b::nat)"

   779   apply (cases "b = 0", simp)

   780   apply (cases "c = 0", simp)

   781   apply (auto simp add: mult_commute divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN mod_eq])

   782   done

   783

   784

   785 subsubsection{*Further Facts about Quotient and Remainder*}

   786

   787 lemma div_1 [simp]: "m div Suc 0 = m"

   788 by (induct m) (simp_all add: div_geq)

   789

   790

   791 (* Monotonicity of div in first argument *)

   792 lemma div_le_mono [rule_format (no_asm)]:

   793     "\<forall>m::nat. m \<le> n --> (m div k) \<le> (n div k)"

   794 apply (case_tac "k=0", simp)

   795 apply (induct "n" rule: nat_less_induct, clarify)

   796 apply (case_tac "n<k")

   797 (* 1  case n<k *)

   798 apply simp

   799 (* 2  case n >= k *)

   800 apply (case_tac "m<k")

   801 (* 2.1  case m<k *)

   802 apply simp

   803 (* 2.2  case m>=k *)

   804 apply (simp add: div_geq diff_le_mono)

   805 done

   806

   807 (* Antimonotonicity of div in second argument *)

   808 lemma div_le_mono2: "!!m::nat. [| 0<m; m\<le>n |] ==> (k div n) \<le> (k div m)"

   809 apply (subgoal_tac "0<n")

   810  prefer 2 apply simp

   811 apply (induct_tac k rule: nat_less_induct)

   812 apply (rename_tac "k")

   813 apply (case_tac "k<n", simp)

   814 apply (subgoal_tac "~ (k<m) ")

   815  prefer 2 apply simp

   816 apply (simp add: div_geq)

   817 apply (subgoal_tac "(k-n) div n \<le> (k-m) div n")

   818  prefer 2

   819  apply (blast intro: div_le_mono diff_le_mono2)

   820 apply (rule le_trans, simp)

   821 apply (simp)

   822 done

   823

   824 lemma div_le_dividend [simp]: "m div n \<le> (m::nat)"

   825 apply (case_tac "n=0", simp)

   826 apply (subgoal_tac "m div n \<le> m div 1", simp)

   827 apply (rule div_le_mono2)

   828 apply (simp_all (no_asm_simp))

   829 done

   830

   831 (* Similar for "less than" *)

   832 lemma div_less_dividend [rule_format]:

   833      "!!n::nat. 1<n ==> 0 < m --> m div n < m"

   834 apply (induct_tac m rule: nat_less_induct)

   835 apply (rename_tac "m")

   836 apply (case_tac "m<n", simp)

   837 apply (subgoal_tac "0<n")

   838  prefer 2 apply simp

   839 apply (simp add: div_geq)

   840 apply (case_tac "n<m")

   841  apply (subgoal_tac "(m-n) div n < (m-n) ")

   842   apply (rule impI less_trans_Suc)+

   843 apply assumption

   844   apply (simp_all)

   845 done

   846

   847 declare div_less_dividend [simp]

   848

   849 text{*A fact for the mutilated chess board*}

   850 lemma mod_Suc: "Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))"

   851 apply (case_tac "n=0", simp)

   852 apply (induct "m" rule: nat_less_induct)

   853 apply (case_tac "Suc (na) <n")

   854 (* case Suc(na) < n *)

   855 apply (frule lessI [THEN less_trans], simp add: less_not_refl3)

   856 (* case n \<le> Suc(na) *)

   857 apply (simp add: linorder_not_less le_Suc_eq mod_geq)

   858 apply (auto simp add: Suc_diff_le le_mod_geq)

   859 done

   860

   861 lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)"

   862 by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)

   863

   864 lemmas mod_eq_0D [dest!] = mod_eq_0_iff [THEN iffD1]

   865

   866 (*Loses information, namely we also have r<d provided d is nonzero*)

   867 lemma mod_eqD: "(m mod d = r) ==> \<exists>q::nat. m = r + q*d"

   868   apply (cut_tac a = m in mod_div_equality)

   869   apply (simp only: add_ac)

   870   apply (blast intro: sym)

   871   done

   872

   873 lemma split_div:

   874  "P(n div k :: nat) =

   875  ((k = 0 \<longrightarrow> P 0) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P i)))"

   876  (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")

   877 proof

   878   assume P: ?P

   879   show ?Q

   880   proof (cases)

   881     assume "k = 0"

   882     with P show ?Q by simp

   883   next

   884     assume not0: "k \<noteq> 0"

   885     thus ?Q

   886     proof (simp, intro allI impI)

   887       fix i j

   888       assume n: "n = k*i + j" and j: "j < k"

   889       show "P i"

   890       proof (cases)

   891         assume "i = 0"

   892         with n j P show "P i" by simp

   893       next

   894         assume "i \<noteq> 0"

   895         with not0 n j P show "P i" by(simp add:add_ac)

   896       qed

   897     qed

   898   qed

   899 next

   900   assume Q: ?Q

   901   show ?P

   902   proof (cases)

   903     assume "k = 0"

   904     with Q show ?P by simp

   905   next

   906     assume not0: "k \<noteq> 0"

   907     with Q have R: ?R by simp

   908     from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]

   909     show ?P by simp

   910   qed

   911 qed

   912

   913 lemma split_div_lemma:

   914   assumes "0 < n"

   915   shows "n * q \<le> m \<and> m < n * Suc q \<longleftrightarrow> q = ((m\<Colon>nat) div n)" (is "?lhs \<longleftrightarrow> ?rhs")

   916 proof

   917   assume ?rhs

   918   with mult_div_cancel have nq: "n * q = m - (m mod n)" by simp

   919   then have A: "n * q \<le> m" by simp

   920   have "n - (m mod n) > 0" using mod_less_divisor assms by auto

   921   then have "m < m + (n - (m mod n))" by simp

   922   then have "m < n + (m - (m mod n))" by simp

   923   with nq have "m < n + n * q" by simp

   924   then have B: "m < n * Suc q" by simp

   925   from A B show ?lhs ..

   926 next

   927   assume P: ?lhs

   928   then have "divmod_nat_rel m n (q, m - n * q)"

   929     unfolding divmod_nat_rel_def by (auto simp add: mult_ac)

   930   with divmod_nat_rel_unique divmod_nat_rel [of m n]

   931   have "(q, m - n * q) = (m div n, m mod n)" by auto

   932   then show ?rhs by simp

   933 qed

   934

   935 theorem split_div':

   936   "P ((m::nat) div n) = ((n = 0 \<and> P 0) \<or>

   937    (\<exists>q. (n * q \<le> m \<and> m < n * (Suc q)) \<and> P q))"

   938   apply (case_tac "0 < n")

   939   apply (simp only: add: split_div_lemma)

   940   apply simp_all

   941   done

   942

   943 lemma split_mod:

   944  "P(n mod k :: nat) =

   945  ((k = 0 \<longrightarrow> P n) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P j)))"

   946  (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")

   947 proof

   948   assume P: ?P

   949   show ?Q

   950   proof (cases)

   951     assume "k = 0"

   952     with P show ?Q by simp

   953   next

   954     assume not0: "k \<noteq> 0"

   955     thus ?Q

   956     proof (simp, intro allI impI)

   957       fix i j

   958       assume "n = k*i + j" "j < k"

   959       thus "P j" using not0 P by(simp add:add_ac mult_ac)

   960     qed

   961   qed

   962 next

   963   assume Q: ?Q

   964   show ?P

   965   proof (cases)

   966     assume "k = 0"

   967     with Q show ?P by simp

   968   next

   969     assume not0: "k \<noteq> 0"

   970     with Q have R: ?R by simp

   971     from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]

   972     show ?P by simp

   973   qed

   974 qed

   975

   976 theorem mod_div_equality': "(m::nat) mod n = m - (m div n) * n"

   977   apply (rule_tac P="%x. m mod n = x - (m div n) * n" in

   978     subst [OF mod_div_equality [of _ n]])

   979   apply arith

   980   done

   981

   982 lemma div_mod_equality':

   983   fixes m n :: nat

   984   shows "m div n * n = m - m mod n"

   985 proof -

   986   have "m mod n \<le> m mod n" ..

   987   from div_mod_equality have

   988     "m div n * n + m mod n - m mod n = m - m mod n" by simp

   989   with diff_add_assoc [OF m mod n \<le> m mod n, of "m div n * n"] have

   990     "m div n * n + (m mod n - m mod n) = m - m mod n"

   991     by simp

   992   then show ?thesis by simp

   993 qed

   994

   995

   996 subsubsection {*An induction'' law for modulus arithmetic.*}

   997

   998 lemma mod_induct_0:

   999   assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"

  1000   and base: "P i" and i: "i<p"

  1001   shows "P 0"

  1002 proof (rule ccontr)

  1003   assume contra: "\<not>(P 0)"

  1004   from i have p: "0<p" by simp

  1005   have "\<forall>k. 0<k \<longrightarrow> \<not> P (p-k)" (is "\<forall>k. ?A k")

  1006   proof

  1007     fix k

  1008     show "?A k"

  1009     proof (induct k)

  1010       show "?A 0" by simp  -- "by contradiction"

  1011     next

  1012       fix n

  1013       assume ih: "?A n"

  1014       show "?A (Suc n)"

  1015       proof (clarsimp)

  1016         assume y: "P (p - Suc n)"

  1017         have n: "Suc n < p"

  1018         proof (rule ccontr)

  1019           assume "\<not>(Suc n < p)"

  1020           hence "p - Suc n = 0"

  1021             by simp

  1022           with y contra show "False"

  1023             by simp

  1024         qed

  1025         hence n2: "Suc (p - Suc n) = p-n" by arith

  1026         from p have "p - Suc n < p" by arith

  1027         with y step have z: "P ((Suc (p - Suc n)) mod p)"

  1028           by blast

  1029         show "False"

  1030         proof (cases "n=0")

  1031           case True

  1032           with z n2 contra show ?thesis by simp

  1033         next

  1034           case False

  1035           with p have "p-n < p" by arith

  1036           with z n2 False ih show ?thesis by simp

  1037         qed

  1038       qed

  1039     qed

  1040   qed

  1041   moreover

  1042   from i obtain k where "0<k \<and> i+k=p"

  1043     by (blast dest: less_imp_add_positive)

  1044   hence "0<k \<and> i=p-k" by auto

  1045   moreover

  1046   note base

  1047   ultimately

  1048   show "False" by blast

  1049 qed

  1050

  1051 lemma mod_induct:

  1052   assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"

  1053   and base: "P i" and i: "i<p" and j: "j<p"

  1054   shows "P j"

  1055 proof -

  1056   have "\<forall>j<p. P j"

  1057   proof

  1058     fix j

  1059     show "j<p \<longrightarrow> P j" (is "?A j")

  1060     proof (induct j)

  1061       from step base i show "?A 0"

  1062         by (auto elim: mod_induct_0)

  1063     next

  1064       fix k

  1065       assume ih: "?A k"

  1066       show "?A (Suc k)"

  1067       proof

  1068         assume suc: "Suc k < p"

  1069         hence k: "k<p" by simp

  1070         with ih have "P k" ..

  1071         with step k have "P (Suc k mod p)"

  1072           by blast

  1073         moreover

  1074         from suc have "Suc k mod p = Suc k"

  1075           by simp

  1076         ultimately

  1077         show "P (Suc k)" by simp

  1078       qed

  1079     qed

  1080   qed

  1081   with j show ?thesis by blast

  1082 qed

  1083

  1084 lemma div2_Suc_Suc [simp]: "Suc (Suc m) div 2 = Suc (m div 2)"

  1085 by (auto simp add: numeral_2_eq_2 le_div_geq)

  1086

  1087 lemma add_self_div_2 [simp]: "(m + m) div 2 = (m::nat)"

  1088 by (simp add: nat_mult_2 [symmetric])

  1089

  1090 lemma mod2_Suc_Suc [simp]: "Suc(Suc(m)) mod 2 = m mod 2"

  1091 apply (subgoal_tac "m mod 2 < 2")

  1092 apply (erule less_2_cases [THEN disjE])

  1093 apply (simp_all (no_asm_simp) add: Let_def mod_Suc nat_1)

  1094 done

  1095

  1096 lemma mod2_gr_0 [simp]: "0 < (m\<Colon>nat) mod 2 \<longleftrightarrow> m mod 2 = 1"

  1097 proof -

  1098   { fix n :: nat have  "(n::nat) < 2 \<Longrightarrow> n = 0 \<or> n = 1" by (induct n) simp_all }

  1099   moreover have "m mod 2 < 2" by simp

  1100   ultimately have "m mod 2 = 0 \<or> m mod 2 = 1" .

  1101   then show ?thesis by auto

  1102 qed

  1103

  1104 text{*These lemmas collapse some needless occurrences of Suc:

  1105     at least three Sucs, since two and fewer are rewritten back to Suc again!

  1106     We already have some rules to simplify operands smaller than 3.*}

  1107

  1108 lemma div_Suc_eq_div_add3 [simp]: "m div (Suc (Suc (Suc n))) = m div (3+n)"

  1109 by (simp add: Suc3_eq_add_3)

  1110

  1111 lemma mod_Suc_eq_mod_add3 [simp]: "m mod (Suc (Suc (Suc n))) = m mod (3+n)"

  1112 by (simp add: Suc3_eq_add_3)

  1113

  1114 lemma Suc_div_eq_add3_div: "(Suc (Suc (Suc m))) div n = (3+m) div n"

  1115 by (simp add: Suc3_eq_add_3)

  1116

  1117 lemma Suc_mod_eq_add3_mod: "(Suc (Suc (Suc m))) mod n = (3+m) mod n"

  1118 by (simp add: Suc3_eq_add_3)

  1119

  1120 lemmas Suc_div_eq_add3_div_number_of =

  1121     Suc_div_eq_add3_div [of _ "number_of v", standard]

  1122 declare Suc_div_eq_add3_div_number_of [simp]

  1123

  1124 lemmas Suc_mod_eq_add3_mod_number_of =

  1125     Suc_mod_eq_add3_mod [of _ "number_of v", standard]

  1126 declare Suc_mod_eq_add3_mod_number_of [simp]

  1127

  1128

  1129 lemma Suc_times_mod_eq: "1<k ==> Suc (k * m) mod k = 1"

  1130 apply (induct "m")

  1131 apply (simp_all add: mod_Suc)

  1132 done

  1133

  1134 declare Suc_times_mod_eq [of "number_of w", standard, simp]

  1135

  1136 lemma [simp]: "n div k \<le> (Suc n) div k"

  1137 by (simp add: div_le_mono)

  1138

  1139 lemma Suc_n_div_2_gt_zero [simp]: "(0::nat) < n ==> 0 < (n + 1) div 2"

  1140 by (cases n) simp_all

  1141

  1142 lemma div_2_gt_zero [simp]: "(1::nat) < n ==> 0 < n div 2"

  1143 using Suc_n_div_2_gt_zero [of "n - 1"] by simp

  1144

  1145   (* Potential use of algebra : Equality modulo n*)

  1146 lemma mod_mult_self3 [simp]: "(k*n + m) mod n = m mod (n::nat)"

  1147 by (simp add: mult_ac add_ac)

  1148

  1149 lemma mod_mult_self4 [simp]: "Suc (k*n + m) mod n = Suc m mod n"

  1150 proof -

  1151   have "Suc (k * n + m) mod n = (k * n + Suc m) mod n" by simp

  1152   also have "... = Suc m mod n" by (rule mod_mult_self3)

  1153   finally show ?thesis .

  1154 qed

  1155

  1156 lemma mod_Suc_eq_Suc_mod: "Suc m mod n = Suc (m mod n) mod n"

  1157 apply (subst mod_Suc [of m])

  1158 apply (subst mod_Suc [of "m mod n"], simp)

  1159 done

  1160

  1161

  1162 subsection {* Division on @{typ int} *}

  1163

  1164 definition divmod_int_rel :: "int \<Rightarrow> int \<Rightarrow> int \<times> int \<Rightarrow> bool" where

  1165     --{*definition of quotient and remainder*}

  1166     [code]: "divmod_int_rel a b = (\<lambda>(q, r). a = b * q + r \<and>

  1167                (if 0 < b then 0 \<le> r \<and> r < b else b < r \<and> r \<le> 0))"

  1168

  1169 definition adjust :: "int \<Rightarrow> int \<times> int \<Rightarrow> int \<times> int" where

  1170     --{*for the division algorithm*}

  1171     [code]: "adjust b = (\<lambda>(q, r). if 0 \<le> r - b then (2 * q + 1, r - b)

  1172                          else (2 * q, r))"

  1173

  1174 text{*algorithm for the case @{text "a\<ge>0, b>0"}*}

  1175 function posDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where

  1176   "posDivAlg a b = (if a < b \<or>  b \<le> 0 then (0, a)

  1177      else adjust b (posDivAlg a (2 * b)))"

  1178 by auto

  1179 termination by (relation "measure (\<lambda>(a, b). nat (a - b + 1))")

  1180   (auto simp add: mult_2)

  1181

  1182 text{*algorithm for the case @{text "a<0, b>0"}*}

  1183 function negDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where

  1184   "negDivAlg a b = (if 0 \<le>a + b \<or> b \<le> 0  then (-1, a + b)

  1185      else adjust b (negDivAlg a (2 * b)))"

  1186 by auto

  1187 termination by (relation "measure (\<lambda>(a, b). nat (- a - b))")

  1188   (auto simp add: mult_2)

  1189

  1190 text{*algorithm for the general case @{term "b\<noteq>0"}*}

  1191 definition negateSnd :: "int \<times> int \<Rightarrow> int \<times> int" where

  1192   [code_unfold]: "negateSnd = apsnd uminus"

  1193

  1194 definition divmod_int :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where

  1195     --{*The full division algorithm considers all possible signs for a, b

  1196        including the special case @{text "a=0, b<0"} because

  1197        @{term negDivAlg} requires @{term "a<0"}.*}

  1198   "divmod_int a b = (if 0 \<le> a then if 0 \<le> b then posDivAlg a b

  1199                   else if a = 0 then (0, 0)

  1200                        else negateSnd (negDivAlg (-a) (-b))

  1201                else

  1202                   if 0 < b then negDivAlg a b

  1203                   else negateSnd (posDivAlg (-a) (-b)))"

  1204

  1205 instantiation int :: Divides.div

  1206 begin

  1207

  1208 definition

  1209   "a div b = fst (divmod_int a b)"

  1210

  1211 definition

  1212  "a mod b = snd (divmod_int a b)"

  1213

  1214 instance ..

  1215

  1216 end

  1217

  1218 lemma divmod_int_mod_div:

  1219   "divmod_int p q = (p div q, p mod q)"

  1220   by (auto simp add: div_int_def mod_int_def)

  1221

  1222 text{*

  1223 Here is the division algorithm in ML:

  1224

  1225 \begin{verbatim}

  1226     fun posDivAlg (a,b) =

  1227       if a<b then (0,a)

  1228       else let val (q,r) = posDivAlg(a, 2*b)

  1229                in  if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)

  1230            end

  1231

  1232     fun negDivAlg (a,b) =

  1233       if 0\<le>a+b then (~1,a+b)

  1234       else let val (q,r) = negDivAlg(a, 2*b)

  1235                in  if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)

  1236            end;

  1237

  1238     fun negateSnd (q,r:int) = (q,~r);

  1239

  1240     fun divmod (a,b) = if 0\<le>a then

  1241                           if b>0 then posDivAlg (a,b)

  1242                            else if a=0 then (0,0)

  1243                                 else negateSnd (negDivAlg (~a,~b))

  1244                        else

  1245                           if 0<b then negDivAlg (a,b)

  1246                           else        negateSnd (posDivAlg (~a,~b));

  1247 \end{verbatim}

  1248 *}

  1249

  1250

  1251

  1252 subsubsection{*Uniqueness and Monotonicity of Quotients and Remainders*}

  1253

  1254 lemma unique_quotient_lemma:

  1255      "[| b*q' + r'  \<le> b*q + r;  0 \<le> r';  r' < b;  r < b |]

  1256       ==> q' \<le> (q::int)"

  1257 apply (subgoal_tac "r' + b * (q'-q) \<le> r")

  1258  prefer 2 apply (simp add: right_diff_distrib)

  1259 apply (subgoal_tac "0 < b * (1 + q - q') ")

  1260 apply (erule_tac [2] order_le_less_trans)

  1261  prefer 2 apply (simp add: right_diff_distrib right_distrib)

  1262 apply (subgoal_tac "b * q' < b * (1 + q) ")

  1263  prefer 2 apply (simp add: right_diff_distrib right_distrib)

  1264 apply (simp add: mult_less_cancel_left)

  1265 done

  1266

  1267 lemma unique_quotient_lemma_neg:

  1268      "[| b*q' + r' \<le> b*q + r;  r \<le> 0;  b < r;  b < r' |]

  1269       ==> q \<le> (q'::int)"

  1270 by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma,

  1271     auto)

  1272

  1273 lemma unique_quotient:

  1274      "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r');  b \<noteq> 0 |]

  1275       ==> q = q'"

  1276 apply (simp add: divmod_int_rel_def linorder_neq_iff split: split_if_asm)

  1277 apply (blast intro: order_antisym

  1278              dest: order_eq_refl [THEN unique_quotient_lemma]

  1279              order_eq_refl [THEN unique_quotient_lemma_neg] sym)+

  1280 done

  1281

  1282

  1283 lemma unique_remainder:

  1284      "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r');  b \<noteq> 0 |]

  1285       ==> r = r'"

  1286 apply (subgoal_tac "q = q'")

  1287  apply (simp add: divmod_int_rel_def)

  1288 apply (blast intro: unique_quotient)

  1289 done

  1290

  1291

  1292 subsubsection{*Correctness of @{term posDivAlg}, the Algorithm for Non-Negative Dividends*}

  1293

  1294 text{*And positive divisors*}

  1295

  1296 lemma adjust_eq [simp]:

  1297      "adjust b (q,r) =

  1298       (let diff = r-b in

  1299         if 0 \<le> diff then (2*q + 1, diff)

  1300                      else (2*q, r))"

  1301 by (simp add: Let_def adjust_def)

  1302

  1303 declare posDivAlg.simps [simp del]

  1304

  1305 text{*use with a simproc to avoid repeatedly proving the premise*}

  1306 lemma posDivAlg_eqn:

  1307      "0 < b ==>

  1308       posDivAlg a b = (if a<b then (0,a) else adjust b (posDivAlg a (2*b)))"

  1309 by (rule posDivAlg.simps [THEN trans], simp)

  1310

  1311 text{*Correctness of @{term posDivAlg}: it computes quotients correctly*}

  1312 theorem posDivAlg_correct:

  1313   assumes "0 \<le> a" and "0 < b"

  1314   shows "divmod_int_rel a b (posDivAlg a b)"

  1315 using prems apply (induct a b rule: posDivAlg.induct)

  1316 apply auto

  1317 apply (simp add: divmod_int_rel_def)

  1318 apply (subst posDivAlg_eqn, simp add: right_distrib)

  1319 apply (case_tac "a < b")

  1320 apply simp_all

  1321 apply (erule splitE)

  1322 apply (auto simp add: right_distrib Let_def mult_ac mult_2_right)

  1323 done

  1324

  1325

  1326 subsubsection{*Correctness of @{term negDivAlg}, the Algorithm for Negative Dividends*}

  1327

  1328 text{*And positive divisors*}

  1329

  1330 declare negDivAlg.simps [simp del]

  1331

  1332 text{*use with a simproc to avoid repeatedly proving the premise*}

  1333 lemma negDivAlg_eqn:

  1334      "0 < b ==>

  1335       negDivAlg a b =

  1336        (if 0\<le>a+b then (-1,a+b) else adjust b (negDivAlg a (2*b)))"

  1337 by (rule negDivAlg.simps [THEN trans], simp)

  1338

  1339 (*Correctness of negDivAlg: it computes quotients correctly

  1340   It doesn't work if a=0 because the 0/b equals 0, not -1*)

  1341 lemma negDivAlg_correct:

  1342   assumes "a < 0" and "b > 0"

  1343   shows "divmod_int_rel a b (negDivAlg a b)"

  1344 using prems apply (induct a b rule: negDivAlg.induct)

  1345 apply (auto simp add: linorder_not_le)

  1346 apply (simp add: divmod_int_rel_def)

  1347 apply (subst negDivAlg_eqn, assumption)

  1348 apply (case_tac "a + b < (0\<Colon>int)")

  1349 apply simp_all

  1350 apply (erule splitE)

  1351 apply (auto simp add: right_distrib Let_def mult_ac mult_2_right)

  1352 done

  1353

  1354

  1355 subsubsection{*Existence Shown by Proving the Division Algorithm to be Correct*}

  1356

  1357 (*the case a=0*)

  1358 lemma divmod_int_rel_0: "b \<noteq> 0 ==> divmod_int_rel 0 b (0, 0)"

  1359 by (auto simp add: divmod_int_rel_def linorder_neq_iff)

  1360

  1361 lemma posDivAlg_0 [simp]: "posDivAlg 0 b = (0, 0)"

  1362 by (subst posDivAlg.simps, auto)

  1363

  1364 lemma negDivAlg_minus1 [simp]: "negDivAlg -1 b = (-1, b - 1)"

  1365 by (subst negDivAlg.simps, auto)

  1366

  1367 lemma negateSnd_eq [simp]: "negateSnd(q,r) = (q,-r)"

  1368 by (simp add: negateSnd_def)

  1369

  1370 lemma divmod_int_rel_neg: "divmod_int_rel (-a) (-b) qr ==> divmod_int_rel a b (negateSnd qr)"

  1371 by (auto simp add: split_ifs divmod_int_rel_def)

  1372

  1373 lemma divmod_int_correct: "b \<noteq> 0 ==> divmod_int_rel a b (divmod_int a b)"

  1374 by (force simp add: linorder_neq_iff divmod_int_rel_0 divmod_int_def divmod_int_rel_neg

  1375                     posDivAlg_correct negDivAlg_correct)

  1376

  1377 text{*Arbitrary definitions for division by zero.  Useful to simplify

  1378     certain equations.*}

  1379

  1380 lemma DIVISION_BY_ZERO [simp]: "a div (0::int) = 0 & a mod (0::int) = a"

  1381 by (simp add: div_int_def mod_int_def divmod_int_def posDivAlg.simps)

  1382

  1383

  1384 text{*Basic laws about division and remainder*}

  1385

  1386 lemma zmod_zdiv_equality: "(a::int) = b * (a div b) + (a mod b)"

  1387 apply (case_tac "b = 0", simp)

  1388 apply (cut_tac a = a and b = b in divmod_int_correct)

  1389 apply (auto simp add: divmod_int_rel_def div_int_def mod_int_def)

  1390 done

  1391

  1392 lemma zdiv_zmod_equality: "(b * (a div b) + (a mod b)) + k = (a::int)+k"

  1393 by(simp add: zmod_zdiv_equality[symmetric])

  1394

  1395 lemma zdiv_zmod_equality2: "((a div b) * b + (a mod b)) + k = (a::int)+k"

  1396 by(simp add: mult_commute zmod_zdiv_equality[symmetric])

  1397

  1398 text {* Tool setup *}

  1399

  1400 ML {*

  1401 local

  1402

  1403 structure CancelDivMod = CancelDivModFun(struct

  1404

  1405   val div_name = @{const_name div};

  1406   val mod_name = @{const_name mod};

  1407   val mk_binop = HOLogic.mk_binop;

  1408   val mk_sum = Arith_Data.mk_sum HOLogic.intT;

  1409   val dest_sum = Arith_Data.dest_sum;

  1410

  1411   val div_mod_eqs = map mk_meta_eq [@{thm zdiv_zmod_equality}, @{thm zdiv_zmod_equality2}];

  1412

  1413   val trans = trans;

  1414

  1415   val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac

  1416     (@{thm diff_minus} :: @{thms add_0s} @ @{thms add_ac}))

  1417

  1418 end)

  1419

  1420 in

  1421

  1422 val cancel_div_mod_int_proc = Simplifier.simproc @{theory}

  1423   "cancel_zdiv_zmod" ["(k::int) + l"] (K CancelDivMod.proc);

  1424

  1425 val _ = Addsimprocs [cancel_div_mod_int_proc];

  1426

  1427 end

  1428 *}

  1429

  1430 lemma pos_mod_conj : "(0::int) < b ==> 0 \<le> a mod b & a mod b < b"

  1431 apply (cut_tac a = a and b = b in divmod_int_correct)

  1432 apply (auto simp add: divmod_int_rel_def mod_int_def)

  1433 done

  1434

  1435 lemmas pos_mod_sign  [simp] = pos_mod_conj [THEN conjunct1, standard]

  1436    and pos_mod_bound [simp] = pos_mod_conj [THEN conjunct2, standard]

  1437

  1438 lemma neg_mod_conj : "b < (0::int) ==> a mod b \<le> 0 & b < a mod b"

  1439 apply (cut_tac a = a and b = b in divmod_int_correct)

  1440 apply (auto simp add: divmod_int_rel_def div_int_def mod_int_def)

  1441 done

  1442

  1443 lemmas neg_mod_sign  [simp] = neg_mod_conj [THEN conjunct1, standard]

  1444    and neg_mod_bound [simp] = neg_mod_conj [THEN conjunct2, standard]

  1445

  1446

  1447

  1448 subsubsection{*General Properties of div and mod*}

  1449

  1450 lemma divmod_int_rel_div_mod: "b \<noteq> 0 ==> divmod_int_rel a b (a div b, a mod b)"

  1451 apply (cut_tac a = a and b = b in zmod_zdiv_equality)

  1452 apply (force simp add: divmod_int_rel_def linorder_neq_iff)

  1453 done

  1454

  1455 lemma divmod_int_rel_div: "[| divmod_int_rel a b (q, r);  b \<noteq> 0 |] ==> a div b = q"

  1456 by (simp add: divmod_int_rel_div_mod [THEN unique_quotient])

  1457

  1458 lemma divmod_int_rel_mod: "[| divmod_int_rel a b (q, r);  b \<noteq> 0 |] ==> a mod b = r"

  1459 by (simp add: divmod_int_rel_div_mod [THEN unique_remainder])

  1460

  1461 lemma div_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a div b = 0"

  1462 apply (rule divmod_int_rel_div)

  1463 apply (auto simp add: divmod_int_rel_def)

  1464 done

  1465

  1466 lemma div_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a div b = 0"

  1467 apply (rule divmod_int_rel_div)

  1468 apply (auto simp add: divmod_int_rel_def)

  1469 done

  1470

  1471 lemma div_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a div b = -1"

  1472 apply (rule divmod_int_rel_div)

  1473 apply (auto simp add: divmod_int_rel_def)

  1474 done

  1475

  1476 (*There is no div_neg_pos_trivial because  0 div b = 0 would supersede it*)

  1477

  1478 lemma mod_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a mod b = a"

  1479 apply (rule_tac q = 0 in divmod_int_rel_mod)

  1480 apply (auto simp add: divmod_int_rel_def)

  1481 done

  1482

  1483 lemma mod_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a mod b = a"

  1484 apply (rule_tac q = 0 in divmod_int_rel_mod)

  1485 apply (auto simp add: divmod_int_rel_def)

  1486 done

  1487

  1488 lemma mod_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a mod b = a+b"

  1489 apply (rule_tac q = "-1" in divmod_int_rel_mod)

  1490 apply (auto simp add: divmod_int_rel_def)

  1491 done

  1492

  1493 text{*There is no @{text mod_neg_pos_trivial}.*}

  1494

  1495

  1496 (*Simpler laws such as -a div b = -(a div b) FAIL, but see just below*)

  1497 lemma zdiv_zminus_zminus [simp]: "(-a) div (-b) = a div (b::int)"

  1498 apply (case_tac "b = 0", simp)

  1499 apply (simp add: divmod_int_rel_div_mod [THEN divmod_int_rel_neg, simplified,

  1500                                  THEN divmod_int_rel_div, THEN sym])

  1501

  1502 done

  1503

  1504 (*Simpler laws such as -a mod b = -(a mod b) FAIL, but see just below*)

  1505 lemma zmod_zminus_zminus [simp]: "(-a) mod (-b) = - (a mod (b::int))"

  1506 apply (case_tac "b = 0", simp)

  1507 apply (subst divmod_int_rel_div_mod [THEN divmod_int_rel_neg, simplified, THEN divmod_int_rel_mod],

  1508        auto)

  1509 done

  1510

  1511

  1512 subsubsection{*Laws for div and mod with Unary Minus*}

  1513

  1514 lemma zminus1_lemma:

  1515      "divmod_int_rel a b (q, r)

  1516       ==> divmod_int_rel (-a) b (if r=0 then -q else -q - 1,

  1517                           if r=0 then 0 else b-r)"

  1518 by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_diff_distrib)

  1519

  1520

  1521 lemma zdiv_zminus1_eq_if:

  1522      "b \<noteq> (0::int)

  1523       ==> (-a) div b =

  1524           (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"

  1525 by (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN divmod_int_rel_div])

  1526

  1527 lemma zmod_zminus1_eq_if:

  1528      "(-a::int) mod b = (if a mod b = 0 then 0 else  b - (a mod b))"

  1529 apply (case_tac "b = 0", simp)

  1530 apply (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN divmod_int_rel_mod])

  1531 done

  1532

  1533 lemma zmod_zminus1_not_zero:

  1534   fixes k l :: int

  1535   shows "- k mod l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"

  1536   unfolding zmod_zminus1_eq_if by auto

  1537

  1538 lemma zdiv_zminus2: "a div (-b) = (-a::int) div b"

  1539 by (cut_tac a = "-a" in zdiv_zminus_zminus, auto)

  1540

  1541 lemma zmod_zminus2: "a mod (-b) = - ((-a::int) mod b)"

  1542 by (cut_tac a = "-a" and b = b in zmod_zminus_zminus, auto)

  1543

  1544 lemma zdiv_zminus2_eq_if:

  1545      "b \<noteq> (0::int)

  1546       ==> a div (-b) =

  1547           (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"

  1548 by (simp add: zdiv_zminus1_eq_if zdiv_zminus2)

  1549

  1550 lemma zmod_zminus2_eq_if:

  1551      "a mod (-b::int) = (if a mod b = 0 then 0 else  (a mod b) - b)"

  1552 by (simp add: zmod_zminus1_eq_if zmod_zminus2)

  1553

  1554 lemma zmod_zminus2_not_zero:

  1555   fixes k l :: int

  1556   shows "k mod - l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"

  1557   unfolding zmod_zminus2_eq_if by auto

  1558

  1559

  1560 subsubsection{*Division of a Number by Itself*}

  1561

  1562 lemma self_quotient_aux1: "[| (0::int) < a; a = r + a*q; r < a |] ==> 1 \<le> q"

  1563 apply (subgoal_tac "0 < a*q")

  1564  apply (simp add: zero_less_mult_iff, arith)

  1565 done

  1566

  1567 lemma self_quotient_aux2: "[| (0::int) < a; a = r + a*q; 0 \<le> r |] ==> q \<le> 1"

  1568 apply (subgoal_tac "0 \<le> a* (1-q) ")

  1569  apply (simp add: zero_le_mult_iff)

  1570 apply (simp add: right_diff_distrib)

  1571 done

  1572

  1573 lemma self_quotient: "[| divmod_int_rel a a (q, r);  a \<noteq> (0::int) |] ==> q = 1"

  1574 apply (simp add: split_ifs divmod_int_rel_def linorder_neq_iff)

  1575 apply (rule order_antisym, safe, simp_all)

  1576 apply (rule_tac [3] a = "-a" and r = "-r" in self_quotient_aux1)

  1577 apply (rule_tac a = "-a" and r = "-r" in self_quotient_aux2)

  1578 apply (force intro: self_quotient_aux1 self_quotient_aux2 simp add: add_commute)+

  1579 done

  1580

  1581 lemma self_remainder: "[| divmod_int_rel a a (q, r);  a \<noteq> (0::int) |] ==> r = 0"

  1582 apply (frule self_quotient, assumption)

  1583 apply (simp add: divmod_int_rel_def)

  1584 done

  1585

  1586 lemma zdiv_self [simp]: "a \<noteq> 0 ==> a div a = (1::int)"

  1587 by (simp add: divmod_int_rel_div_mod [THEN self_quotient])

  1588

  1589 (*Here we have 0 mod 0 = 0, also assumed by Knuth (who puts m mod 0 = 0) *)

  1590 lemma zmod_self [simp]: "a mod a = (0::int)"

  1591 apply (case_tac "a = 0", simp)

  1592 apply (simp add: divmod_int_rel_div_mod [THEN self_remainder])

  1593 done

  1594

  1595

  1596 subsubsection{*Computation of Division and Remainder*}

  1597

  1598 lemma zdiv_zero [simp]: "(0::int) div b = 0"

  1599 by (simp add: div_int_def divmod_int_def)

  1600

  1601 lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"

  1602 by (simp add: div_int_def divmod_int_def)

  1603

  1604 lemma zmod_zero [simp]: "(0::int) mod b = 0"

  1605 by (simp add: mod_int_def divmod_int_def)

  1606

  1607 lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"

  1608 by (simp add: mod_int_def divmod_int_def)

  1609

  1610 text{*a positive, b positive *}

  1611

  1612 lemma div_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a div b = fst (posDivAlg a b)"

  1613 by (simp add: div_int_def divmod_int_def)

  1614

  1615 lemma mod_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a mod b = snd (posDivAlg a b)"

  1616 by (simp add: mod_int_def divmod_int_def)

  1617

  1618 text{*a negative, b positive *}

  1619

  1620 lemma div_neg_pos: "[| a < 0;  0 < b |] ==> a div b = fst (negDivAlg a b)"

  1621 by (simp add: div_int_def divmod_int_def)

  1622

  1623 lemma mod_neg_pos: "[| a < 0;  0 < b |] ==> a mod b = snd (negDivAlg a b)"

  1624 by (simp add: mod_int_def divmod_int_def)

  1625

  1626 text{*a positive, b negative *}

  1627

  1628 lemma div_pos_neg:

  1629      "[| 0 < a;  b < 0 |] ==> a div b = fst (negateSnd (negDivAlg (-a) (-b)))"

  1630 by (simp add: div_int_def divmod_int_def)

  1631

  1632 lemma mod_pos_neg:

  1633      "[| 0 < a;  b < 0 |] ==> a mod b = snd (negateSnd (negDivAlg (-a) (-b)))"

  1634 by (simp add: mod_int_def divmod_int_def)

  1635

  1636 text{*a negative, b negative *}

  1637

  1638 lemma div_neg_neg:

  1639      "[| a < 0;  b \<le> 0 |] ==> a div b = fst (negateSnd (posDivAlg (-a) (-b)))"

  1640 by (simp add: div_int_def divmod_int_def)

  1641

  1642 lemma mod_neg_neg:

  1643      "[| a < 0;  b \<le> 0 |] ==> a mod b = snd (negateSnd (posDivAlg (-a) (-b)))"

  1644 by (simp add: mod_int_def divmod_int_def)

  1645

  1646 text {*Simplify expresions in which div and mod combine numerical constants*}

  1647

  1648 lemma divmod_int_relI:

  1649   "\<lbrakk>a == b * q + r; if 0 < b then 0 \<le> r \<and> r < b else b < r \<and> r \<le> 0\<rbrakk>

  1650     \<Longrightarrow> divmod_int_rel a b (q, r)"

  1651   unfolding divmod_int_rel_def by simp

  1652

  1653 lemmas divmod_int_rel_div_eq = divmod_int_relI [THEN divmod_int_rel_div, THEN eq_reflection]

  1654 lemmas divmod_int_rel_mod_eq = divmod_int_relI [THEN divmod_int_rel_mod, THEN eq_reflection]

  1655 lemmas arithmetic_simps =

  1656   arith_simps

  1657   add_special

  1658   OrderedGroup.add_0_left

  1659   OrderedGroup.add_0_right

  1660   mult_zero_left

  1661   mult_zero_right

  1662   mult_1_left

  1663   mult_1_right

  1664

  1665 (* simprocs adapted from HOL/ex/Binary.thy *)

  1666 ML {*

  1667 local

  1668   val mk_number = HOLogic.mk_number HOLogic.intT;

  1669   fun mk_cert u k l = @{term "plus :: int \<Rightarrow> int \<Rightarrow> int"} $  1670 (@{term "times :: int \<Rightarrow> int \<Rightarrow> int"}$ u $mk_number k)$

  1671       mk_number l;

  1672   fun prove ctxt prop = Goal.prove ctxt [] [] prop

  1673     (K (ALLGOALS (full_simp_tac (HOL_basic_ss addsimps @{thms arithmetic_simps}))));

  1674   fun binary_proc proc ss ct =

  1675     (case Thm.term_of ct of

  1676       _ $t$ u =>

  1677       (case try (pairself ((snd o HOLogic.dest_number))) (t, u) of

  1678         SOME args => proc (Simplifier.the_context ss) args

  1679       | NONE => NONE)

  1680     | _ => NONE);

  1681 in

  1682   fun divmod_proc rule = binary_proc (fn ctxt => fn ((m, t), (n, u)) =>

  1683     if n = 0 then NONE

  1684     else let val (k, l) = Integer.div_mod m n;

  1685     in SOME (rule OF [prove ctxt (Logic.mk_equals (t, mk_cert u k l))]) end);

  1686 end

  1687 *}

  1688

  1689 simproc_setup binary_int_div ("number_of m div number_of n :: int") =

  1690   {* K (divmod_proc (@{thm divmod_int_rel_div_eq})) *}

  1691

  1692 simproc_setup binary_int_mod ("number_of m mod number_of n :: int") =

  1693   {* K (divmod_proc (@{thm divmod_int_rel_mod_eq})) *}

  1694

  1695 lemmas posDivAlg_eqn_number_of [simp] =

  1696     posDivAlg_eqn [of "number_of v" "number_of w", standard]

  1697

  1698 lemmas negDivAlg_eqn_number_of [simp] =

  1699     negDivAlg_eqn [of "number_of v" "number_of w", standard]

  1700

  1701

  1702 text{*Special-case simplification *}

  1703

  1704 lemma zmod_minus1_right [simp]: "a mod (-1::int) = 0"

  1705 apply (cut_tac a = a and b = "-1" in neg_mod_sign)

  1706 apply (cut_tac [2] a = a and b = "-1" in neg_mod_bound)

  1707 apply (auto simp del: neg_mod_sign neg_mod_bound)

  1708 done

  1709

  1710 lemma zdiv_minus1_right [simp]: "a div (-1::int) = -a"

  1711 by (cut_tac a = a and b = "-1" in zmod_zdiv_equality, auto)

  1712

  1713 (** The last remaining special cases for constant arithmetic:

  1714     1 div z and 1 mod z **)

  1715

  1716 lemmas div_pos_pos_1_number_of [simp] =

  1717     div_pos_pos [OF int_0_less_1, of "number_of w", standard]

  1718

  1719 lemmas div_pos_neg_1_number_of [simp] =

  1720     div_pos_neg [OF int_0_less_1, of "number_of w", standard]

  1721

  1722 lemmas mod_pos_pos_1_number_of [simp] =

  1723     mod_pos_pos [OF int_0_less_1, of "number_of w", standard]

  1724

  1725 lemmas mod_pos_neg_1_number_of [simp] =

  1726     mod_pos_neg [OF int_0_less_1, of "number_of w", standard]

  1727

  1728

  1729 lemmas posDivAlg_eqn_1_number_of [simp] =

  1730     posDivAlg_eqn [of concl: 1 "number_of w", standard]

  1731

  1732 lemmas negDivAlg_eqn_1_number_of [simp] =

  1733     negDivAlg_eqn [of concl: 1 "number_of w", standard]

  1734

  1735

  1736

  1737 subsubsection{*Monotonicity in the First Argument (Dividend)*}

  1738

  1739 lemma zdiv_mono1: "[| a \<le> a';  0 < (b::int) |] ==> a div b \<le> a' div b"

  1740 apply (cut_tac a = a and b = b in zmod_zdiv_equality)

  1741 apply (cut_tac a = a' and b = b in zmod_zdiv_equality)

  1742 apply (rule unique_quotient_lemma)

  1743 apply (erule subst)

  1744 apply (erule subst, simp_all)

  1745 done

  1746

  1747 lemma zdiv_mono1_neg: "[| a \<le> a';  (b::int) < 0 |] ==> a' div b \<le> a div b"

  1748 apply (cut_tac a = a and b = b in zmod_zdiv_equality)

  1749 apply (cut_tac a = a' and b = b in zmod_zdiv_equality)

  1750 apply (rule unique_quotient_lemma_neg)

  1751 apply (erule subst)

  1752 apply (erule subst, simp_all)

  1753 done

  1754

  1755

  1756 subsubsection{*Monotonicity in the Second Argument (Divisor)*}

  1757

  1758 lemma q_pos_lemma:

  1759      "[| 0 \<le> b'*q' + r'; r' < b';  0 < b' |] ==> 0 \<le> (q'::int)"

  1760 apply (subgoal_tac "0 < b'* (q' + 1) ")

  1761  apply (simp add: zero_less_mult_iff)

  1762 apply (simp add: right_distrib)

  1763 done

  1764

  1765 lemma zdiv_mono2_lemma:

  1766      "[| b*q + r = b'*q' + r';  0 \<le> b'*q' + r';

  1767          r' < b';  0 \<le> r;  0 < b';  b' \<le> b |]

  1768       ==> q \<le> (q'::int)"

  1769 apply (frule q_pos_lemma, assumption+)

  1770 apply (subgoal_tac "b*q < b* (q' + 1) ")

  1771  apply (simp add: mult_less_cancel_left)

  1772 apply (subgoal_tac "b*q = r' - r + b'*q'")

  1773  prefer 2 apply simp

  1774 apply (simp (no_asm_simp) add: right_distrib)

  1775 apply (subst add_commute, rule zadd_zless_mono, arith)

  1776 apply (rule mult_right_mono, auto)

  1777 done

  1778

  1779 lemma zdiv_mono2:

  1780      "[| (0::int) \<le> a;  0 < b';  b' \<le> b |] ==> a div b \<le> a div b'"

  1781 apply (subgoal_tac "b \<noteq> 0")

  1782  prefer 2 apply arith

  1783 apply (cut_tac a = a and b = b in zmod_zdiv_equality)

  1784 apply (cut_tac a = a and b = b' in zmod_zdiv_equality)

  1785 apply (rule zdiv_mono2_lemma)

  1786 apply (erule subst)

  1787 apply (erule subst, simp_all)

  1788 done

  1789

  1790 lemma q_neg_lemma:

  1791      "[| b'*q' + r' < 0;  0 \<le> r';  0 < b' |] ==> q' \<le> (0::int)"

  1792 apply (subgoal_tac "b'*q' < 0")

  1793  apply (simp add: mult_less_0_iff, arith)

  1794 done

  1795

  1796 lemma zdiv_mono2_neg_lemma:

  1797      "[| b*q + r = b'*q' + r';  b'*q' + r' < 0;

  1798          r < b;  0 \<le> r';  0 < b';  b' \<le> b |]

  1799       ==> q' \<le> (q::int)"

  1800 apply (frule q_neg_lemma, assumption+)

  1801 apply (subgoal_tac "b*q' < b* (q + 1) ")

  1802  apply (simp add: mult_less_cancel_left)

  1803 apply (simp add: right_distrib)

  1804 apply (subgoal_tac "b*q' \<le> b'*q'")

  1805  prefer 2 apply (simp add: mult_right_mono_neg, arith)

  1806 done

  1807

  1808 lemma zdiv_mono2_neg:

  1809      "[| a < (0::int);  0 < b';  b' \<le> b |] ==> a div b' \<le> a div b"

  1810 apply (cut_tac a = a and b = b in zmod_zdiv_equality)

  1811 apply (cut_tac a = a and b = b' in zmod_zdiv_equality)

  1812 apply (rule zdiv_mono2_neg_lemma)

  1813 apply (erule subst)

  1814 apply (erule subst, simp_all)

  1815 done

  1816

  1817

  1818 subsubsection{*More Algebraic Laws for div and mod*}

  1819

  1820 text{*proving (a*b) div c = a * (b div c) + a * (b mod c) *}

  1821

  1822 lemma zmult1_lemma:

  1823      "[| divmod_int_rel b c (q, r);  c \<noteq> 0 |]

  1824       ==> divmod_int_rel (a * b) c (a*q + a*r div c, a*r mod c)"

  1825 by (auto simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_distrib mult_ac)

  1826

  1827 lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"

  1828 apply (case_tac "c = 0", simp)

  1829 apply (blast intro: divmod_int_rel_div_mod [THEN zmult1_lemma, THEN divmod_int_rel_div])

  1830 done

  1831

  1832 lemma zmod_zmult1_eq: "(a*b) mod c = a*(b mod c) mod (c::int)"

  1833 apply (case_tac "c = 0", simp)

  1834 apply (blast intro: divmod_int_rel_div_mod [THEN zmult1_lemma, THEN divmod_int_rel_mod])

  1835 done

  1836

  1837 lemma zmod_zdiv_trivial: "(a mod b) div b = (0::int)"

  1838 apply (case_tac "b = 0", simp)

  1839 apply (auto simp add: linorder_neq_iff div_pos_pos_trivial div_neg_neg_trivial)

  1840 done

  1841

  1842 text{*proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) *}

  1843

  1844 lemma zadd1_lemma:

  1845      "[| divmod_int_rel a c (aq, ar);  divmod_int_rel b c (bq, br);  c \<noteq> 0 |]

  1846       ==> divmod_int_rel (a+b) c (aq + bq + (ar+br) div c, (ar+br) mod c)"

  1847 by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_distrib)

  1848

  1849 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)

  1850 lemma zdiv_zadd1_eq:

  1851      "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"

  1852 apply (case_tac "c = 0", simp)

  1853 apply (blast intro: zadd1_lemma [OF divmod_int_rel_div_mod divmod_int_rel_div_mod] divmod_int_rel_div)

  1854 done

  1855

  1856 instance int :: ring_div

  1857 proof

  1858   fix a b c :: int

  1859   assume not0: "b \<noteq> 0"

  1860   show "(a + c * b) div b = c + a div b"

  1861     unfolding zdiv_zadd1_eq [of a "c * b"] using not0

  1862       by (simp add: zmod_zmult1_eq zmod_zdiv_trivial zdiv_zmult1_eq)

  1863 next

  1864   fix a b c :: int

  1865   assume "a \<noteq> 0"

  1866   then show "(a * b) div (a * c) = b div c"

  1867   proof (cases "b \<noteq> 0 \<and> c \<noteq> 0")

  1868     case False then show ?thesis by auto

  1869   next

  1870     case True then have "b \<noteq> 0" and "c \<noteq> 0" by auto

  1871     with a \<noteq> 0

  1872     have "\<And>q r. divmod_int_rel b c (q, r) \<Longrightarrow> divmod_int_rel (a * b) (a * c) (q, a * r)"

  1873       apply (auto simp add: divmod_int_rel_def)

  1874       apply (auto simp add: algebra_simps)

  1875       apply (auto simp add: zero_less_mult_iff zero_le_mult_iff mult_le_0_iff mult_commute [of a] mult_less_cancel_right)

  1876       done

  1877     moreover with c \<noteq> 0 divmod_int_rel_div_mod have "divmod_int_rel b c (b div c, b mod c)" by auto

  1878     ultimately have "divmod_int_rel (a * b) (a * c) (b div c, a * (b mod c))" .

  1879     moreover from  a \<noteq> 0 c \<noteq> 0 have "a * c \<noteq> 0" by simp

  1880     ultimately show ?thesis by (rule divmod_int_rel_div)

  1881   qed

  1882 qed auto

  1883

  1884 lemma posDivAlg_div_mod:

  1885   assumes "k \<ge> 0"

  1886   and "l \<ge> 0"

  1887   shows "posDivAlg k l = (k div l, k mod l)"

  1888 proof (cases "l = 0")

  1889   case True then show ?thesis by (simp add: posDivAlg.simps)

  1890 next

  1891   case False with assms posDivAlg_correct

  1892     have "divmod_int_rel k l (fst (posDivAlg k l), snd (posDivAlg k l))"

  1893     by simp

  1894   from divmod_int_rel_div [OF this l \<noteq> 0] divmod_int_rel_mod [OF this l \<noteq> 0]

  1895   show ?thesis by simp

  1896 qed

  1897

  1898 lemma negDivAlg_div_mod:

  1899   assumes "k < 0"

  1900   and "l > 0"

  1901   shows "negDivAlg k l = (k div l, k mod l)"

  1902 proof -

  1903   from assms have "l \<noteq> 0" by simp

  1904   from assms negDivAlg_correct

  1905     have "divmod_int_rel k l (fst (negDivAlg k l), snd (negDivAlg k l))"

  1906     by simp

  1907   from divmod_int_rel_div [OF this l \<noteq> 0] divmod_int_rel_mod [OF this l \<noteq> 0]

  1908   show ?thesis by simp

  1909 qed

  1910

  1911 lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"

  1912 by (simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)

  1913

  1914 (* REVISIT: should this be generalized to all semiring_div types? *)

  1915 lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]

  1916

  1917

  1918 subsubsection{*Proving  @{term "a div (b*c) = (a div b) div c"} *}

  1919

  1920 (*The condition c>0 seems necessary.  Consider that 7 div ~6 = ~2 but

  1921   7 div 2 div ~3 = 3 div ~3 = ~1.  The subcase (a div b) mod c = 0 seems

  1922   to cause particular problems.*)

  1923

  1924 text{*first, four lemmas to bound the remainder for the cases b<0 and b>0 *}

  1925

  1926 lemma zmult2_lemma_aux1: "[| (0::int) < c;  b < r;  r \<le> 0 |] ==> b*c < b*(q mod c) + r"

  1927 apply (subgoal_tac "b * (c - q mod c) < r * 1")

  1928  apply (simp add: algebra_simps)

  1929 apply (rule order_le_less_trans)

  1930  apply (erule_tac [2] mult_strict_right_mono)

  1931  apply (rule mult_left_mono_neg)

  1932   using add1_zle_eq[of "q mod c"]apply(simp add: algebra_simps pos_mod_bound)

  1933  apply (simp)

  1934 apply (simp)

  1935 done

  1936

  1937 lemma zmult2_lemma_aux2:

  1938      "[| (0::int) < c;   b < r;  r \<le> 0 |] ==> b * (q mod c) + r \<le> 0"

  1939 apply (subgoal_tac "b * (q mod c) \<le> 0")

  1940  apply arith

  1941 apply (simp add: mult_le_0_iff)

  1942 done

  1943

  1944 lemma zmult2_lemma_aux3: "[| (0::int) < c;  0 \<le> r;  r < b |] ==> 0 \<le> b * (q mod c) + r"

  1945 apply (subgoal_tac "0 \<le> b * (q mod c) ")

  1946 apply arith

  1947 apply (simp add: zero_le_mult_iff)

  1948 done

  1949

  1950 lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \<le> r; r < b |] ==> b * (q mod c) + r < b * c"

  1951 apply (subgoal_tac "r * 1 < b * (c - q mod c) ")

  1952  apply (simp add: right_diff_distrib)

  1953 apply (rule order_less_le_trans)

  1954  apply (erule mult_strict_right_mono)

  1955  apply (rule_tac [2] mult_left_mono)

  1956   apply simp

  1957  using add1_zle_eq[of "q mod c"] apply (simp add: algebra_simps pos_mod_bound)

  1958 apply simp

  1959 done

  1960

  1961 lemma zmult2_lemma: "[| divmod_int_rel a b (q, r);  b \<noteq> 0;  0 < c |]

  1962       ==> divmod_int_rel a (b * c) (q div c, b*(q mod c) + r)"

  1963 by (auto simp add: mult_ac divmod_int_rel_def linorder_neq_iff

  1964                    zero_less_mult_iff right_distrib [symmetric]

  1965                    zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4)

  1966

  1967 lemma zdiv_zmult2_eq: "(0::int) < c ==> a div (b*c) = (a div b) div c"

  1968 apply (case_tac "b = 0", simp)

  1969 apply (force simp add: divmod_int_rel_div_mod [THEN zmult2_lemma, THEN divmod_int_rel_div])

  1970 done

  1971

  1972 lemma zmod_zmult2_eq:

  1973      "(0::int) < c ==> a mod (b*c) = b*(a div b mod c) + a mod b"

  1974 apply (case_tac "b = 0", simp)

  1975 apply (force simp add: divmod_int_rel_div_mod [THEN zmult2_lemma, THEN divmod_int_rel_mod])

  1976 done

  1977

  1978

  1979 subsubsection {*Splitting Rules for div and mod*}

  1980

  1981 text{*The proofs of the two lemmas below are essentially identical*}

  1982

  1983 lemma split_pos_lemma:

  1984  "0<k ==>

  1985     P(n div k :: int)(n mod k) = (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i j)"

  1986 apply (rule iffI, clarify)

  1987  apply (erule_tac P="P ?x ?y" in rev_mp)

  1988  apply (subst mod_add_eq)

  1989  apply (subst zdiv_zadd1_eq)

  1990  apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)

  1991 txt{*converse direction*}

  1992 apply (drule_tac x = "n div k" in spec)

  1993 apply (drule_tac x = "n mod k" in spec, simp)

  1994 done

  1995

  1996 lemma split_neg_lemma:

  1997  "k<0 ==>

  1998     P(n div k :: int)(n mod k) = (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i j)"

  1999 apply (rule iffI, clarify)

  2000  apply (erule_tac P="P ?x ?y" in rev_mp)

  2001  apply (subst mod_add_eq)

  2002  apply (subst zdiv_zadd1_eq)

  2003  apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)

  2004 txt{*converse direction*}

  2005 apply (drule_tac x = "n div k" in spec)

  2006 apply (drule_tac x = "n mod k" in spec, simp)

  2007 done

  2008

  2009 lemma split_zdiv:

  2010  "P(n div k :: int) =

  2011   ((k = 0 --> P 0) &

  2012    (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i)) &

  2013    (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i)))"

  2014 apply (case_tac "k=0", simp)

  2015 apply (simp only: linorder_neq_iff)

  2016 apply (erule disjE)

  2017  apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"]

  2018                       split_neg_lemma [of concl: "%x y. P x"])

  2019 done

  2020

  2021 lemma split_zmod:

  2022  "P(n mod k :: int) =

  2023   ((k = 0 --> P n) &

  2024    (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P j)) &

  2025    (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P j)))"

  2026 apply (case_tac "k=0", simp)

  2027 apply (simp only: linorder_neq_iff)

  2028 apply (erule disjE)

  2029  apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"]

  2030                       split_neg_lemma [of concl: "%x y. P y"])

  2031 done

  2032

  2033 (* Enable arith to deal with @{term div} and @{term mod} when

  2034    these are applied to some constant that is of the form

  2035    @{term "number_of k"}: *)

  2036 declare split_zdiv [of _ _ "number_of k", standard, arith_split]

  2037 declare split_zmod [of _ _ "number_of k", standard, arith_split]

  2038

  2039

  2040 subsubsection{*Speeding up the Division Algorithm with Shifting*}

  2041

  2042 text{*computing div by shifting *}

  2043

  2044 lemma pos_zdiv_mult_2: "(0::int) \<le> a ==> (1 + 2*b) div (2*a) = b div a"

  2045 proof cases

  2046   assume "a=0"

  2047     thus ?thesis by simp

  2048 next

  2049   assume "a\<noteq>0" and le_a: "0\<le>a"

  2050   hence a_pos: "1 \<le> a" by arith

  2051   hence one_less_a2: "1 < 2 * a" by arith

  2052   hence le_2a: "2 * (1 + b mod a) \<le> 2 * a"

  2053     unfolding mult_le_cancel_left

  2054     by (simp add: add1_zle_eq add_commute [of 1])

  2055   with a_pos have "0 \<le> b mod a" by simp

  2056   hence le_addm: "0 \<le> 1 mod (2*a) + 2*(b mod a)"

  2057     by (simp add: mod_pos_pos_trivial one_less_a2)

  2058   with  le_2a

  2059   have "(1 mod (2*a) + 2*(b mod a)) div (2*a) = 0"

  2060     by (simp add: div_pos_pos_trivial le_addm mod_pos_pos_trivial one_less_a2

  2061                   right_distrib)

  2062   thus ?thesis

  2063     by (subst zdiv_zadd1_eq,

  2064         simp add: mod_mult_mult1 one_less_a2

  2065                   div_pos_pos_trivial)

  2066 qed

  2067

  2068 lemma neg_zdiv_mult_2: "a \<le> (0::int) ==> (1 + 2*b) div (2*a) = (b+1) div a"

  2069 apply (subgoal_tac " (1 + 2* (-b - 1)) div (2 * (-a)) = (-b - 1) div (-a) ")

  2070 apply (rule_tac [2] pos_zdiv_mult_2)

  2071 apply (auto simp add: right_diff_distrib)

  2072 apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))")

  2073 apply (simp only: zdiv_zminus_zminus diff_minus minus_add_distrib [symmetric])

  2074 apply (simp_all add: algebra_simps)

  2075 apply (simp only: ab_diff_minus minus_add_distrib [symmetric] number_of_Min zdiv_zminus_zminus)

  2076 done

  2077

  2078 lemma zdiv_number_of_Bit0 [simp]:

  2079      "number_of (Int.Bit0 v) div number_of (Int.Bit0 w) =

  2080           number_of v div (number_of w :: int)"

  2081 by (simp only: number_of_eq numeral_simps) (simp add: mult_2 [symmetric])

  2082

  2083 lemma zdiv_number_of_Bit1 [simp]:

  2084      "number_of (Int.Bit1 v) div number_of (Int.Bit0 w) =

  2085           (if (0::int) \<le> number_of w

  2086            then number_of v div (number_of w)

  2087            else (number_of v + (1::int)) div (number_of w))"

  2088 apply (simp only: number_of_eq numeral_simps UNIV_I split: split_if)

  2089 apply (simp add: pos_zdiv_mult_2 neg_zdiv_mult_2 add_ac mult_2 [symmetric])

  2090 done

  2091

  2092

  2093 subsubsection{*Computing mod by Shifting (proofs resemble those for div)*}

  2094

  2095 lemma pos_zmod_mult_2:

  2096   fixes a b :: int

  2097   assumes "0 \<le> a"

  2098   shows "(1 + 2 * b) mod (2 * a) = 1 + 2 * (b mod a)"

  2099 proof (cases "0 < a")

  2100   case False with assms show ?thesis by simp

  2101 next

  2102   case True

  2103   then have "b mod a < a" by (rule pos_mod_bound)

  2104   then have "1 + b mod a \<le> a" by simp

  2105   then have A: "2 * (1 + b mod a) \<le> 2 * a" by simp

  2106   from 0 < a have "0 \<le> b mod a" by (rule pos_mod_sign)

  2107   then have B: "0 \<le> 1 + 2 * (b mod a)" by simp

  2108   have "((1\<Colon>int) mod ((2\<Colon>int) * a) + (2\<Colon>int) * b mod ((2\<Colon>int) * a)) mod ((2\<Colon>int) * a) = (1\<Colon>int) + (2\<Colon>int) * (b mod a)"

  2109     using 0 < a and A

  2110     by (auto simp add: mod_mult_mult1 mod_pos_pos_trivial ring_distribs intro!: mod_pos_pos_trivial B)

  2111   then show ?thesis by (subst mod_add_eq)

  2112 qed

  2113

  2114 lemma neg_zmod_mult_2:

  2115   fixes a b :: int

  2116   assumes "a \<le> 0"

  2117   shows "(1 + 2 * b) mod (2 * a) = 2 * ((b + 1) mod a) - 1"

  2118 proof -

  2119   from assms have "0 \<le> - a" by auto

  2120   then have "(1 + 2 * (- b - 1)) mod (2 * (- a)) = 1 + 2 * ((- b - 1) mod (- a))"

  2121     by (rule pos_zmod_mult_2)

  2122   then show ?thesis by (simp add: zmod_zminus2 algebra_simps)

  2123      (simp add: diff_minus add_ac)

  2124 qed

  2125

  2126 lemma zmod_number_of_Bit0 [simp]:

  2127      "number_of (Int.Bit0 v) mod number_of (Int.Bit0 w) =

  2128       (2::int) * (number_of v mod number_of w)"

  2129 apply (simp only: number_of_eq numeral_simps)

  2130 apply (simp add: mod_mult_mult1 pos_zmod_mult_2

  2131                  neg_zmod_mult_2 add_ac mult_2 [symmetric])

  2132 done

  2133

  2134 lemma zmod_number_of_Bit1 [simp]:

  2135      "number_of (Int.Bit1 v) mod number_of (Int.Bit0 w) =

  2136       (if (0::int) \<le> number_of w

  2137                 then 2 * (number_of v mod number_of w) + 1

  2138                 else 2 * ((number_of v + (1::int)) mod number_of w) - 1)"

  2139 apply (simp only: number_of_eq numeral_simps)

  2140 apply (simp add: mod_mult_mult1 pos_zmod_mult_2

  2141                  neg_zmod_mult_2 add_ac mult_2 [symmetric])

  2142 done

  2143

  2144

  2145 subsubsection{*Quotients of Signs*}

  2146

  2147 lemma div_neg_pos_less0: "[| a < (0::int);  0 < b |] ==> a div b < 0"

  2148 apply (subgoal_tac "a div b \<le> -1", force)

  2149 apply (rule order_trans)

  2150 apply (rule_tac a' = "-1" in zdiv_mono1)

  2151 apply (auto simp add: div_eq_minus1)

  2152 done

  2153

  2154 lemma div_nonneg_neg_le0: "[| (0::int) \<le> a; b < 0 |] ==> a div b \<le> 0"

  2155 by (drule zdiv_mono1_neg, auto)

  2156

  2157 lemma div_nonpos_pos_le0: "[| (a::int) \<le> 0; b > 0 |] ==> a div b \<le> 0"

  2158 by (drule zdiv_mono1, auto)

  2159

  2160 lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \<le> a div b) = (0 \<le> a)"

  2161 apply auto

  2162 apply (drule_tac [2] zdiv_mono1)

  2163 apply (auto simp add: linorder_neq_iff)

  2164 apply (simp (no_asm_use) add: linorder_not_less [symmetric])

  2165 apply (blast intro: div_neg_pos_less0)

  2166 done

  2167

  2168 lemma neg_imp_zdiv_nonneg_iff:

  2169      "b < (0::int) ==> (0 \<le> a div b) = (a \<le> (0::int))"

  2170 apply (subst zdiv_zminus_zminus [symmetric])

  2171 apply (subst pos_imp_zdiv_nonneg_iff, auto)

  2172 done

  2173

  2174 (*But not (a div b \<le> 0 iff a\<le>0); consider a=1, b=2 when a div b = 0.*)

  2175 lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"

  2176 by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)

  2177

  2178 (*Again the law fails for \<le>: consider a = -1, b = -2 when a div b = 0*)

  2179 lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"

  2180 by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)

  2181

  2182

  2183 subsubsection {* The Divides Relation *}

  2184

  2185 lemmas zdvd_iff_zmod_eq_0_number_of [simp] =

  2186   dvd_eq_mod_eq_0 [of "number_of x::int" "number_of y::int", standard]

  2187

  2188 lemma zdvd_zmod: "f dvd m ==> f dvd (n::int) ==> f dvd m mod n"

  2189   by (rule dvd_mod) (* TODO: remove *)

  2190

  2191 lemma zdvd_zmod_imp_zdvd: "k dvd m mod n ==> k dvd n ==> k dvd (m::int)"

  2192   by (rule dvd_mod_imp_dvd) (* TODO: remove *)

  2193

  2194 lemma zmult_div_cancel: "(n::int) * (m div n) = m - (m mod n)"

  2195   using zmod_zdiv_equality[where a="m" and b="n"]

  2196   by (simp add: algebra_simps)

  2197

  2198 lemma zpower_zmod: "((x::int) mod m)^y mod m = x^y mod m"

  2199 apply (induct "y", auto)

  2200 apply (rule zmod_zmult1_eq [THEN trans])

  2201 apply (simp (no_asm_simp))

  2202 apply (rule mod_mult_eq [symmetric])

  2203 done

  2204

  2205 lemma zdiv_int: "int (a div b) = (int a) div (int b)"

  2206 apply (subst split_div, auto)

  2207 apply (subst split_zdiv, auto)

  2208 apply (rule_tac a="int (b * i) + int j" and b="int b" and r="int j" and r'=ja in unique_quotient)

  2209 apply (auto simp add: divmod_int_rel_def of_nat_mult)

  2210 done

  2211

  2212 lemma zmod_int: "int (a mod b) = (int a) mod (int b)"

  2213 apply (subst split_mod, auto)

  2214 apply (subst split_zmod, auto)

  2215 apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia

  2216        in unique_remainder)

  2217 apply (auto simp add: divmod_int_rel_def of_nat_mult)

  2218 done

  2219

  2220 lemma abs_div: "(y::int) dvd x \<Longrightarrow> abs (x div y) = abs x div abs y"

  2221 by (unfold dvd_def, cases "y=0", auto simp add: abs_mult)

  2222

  2223 lemma zdvd_mult_div_cancel:"(n::int) dvd m \<Longrightarrow> n * (m div n) = m"

  2224 apply (subgoal_tac "m mod n = 0")

  2225  apply (simp add: zmult_div_cancel)

  2226 apply (simp only: dvd_eq_mod_eq_0)

  2227 done

  2228

  2229 text{*Suggested by Matthias Daum*}

  2230 lemma int_power_div_base:

  2231      "\<lbrakk>0 < m; 0 < k\<rbrakk> \<Longrightarrow> k ^ m div k = (k::int) ^ (m - Suc 0)"

  2232 apply (subgoal_tac "k ^ m = k ^ ((m - Suc 0) + Suc 0)")

  2233  apply (erule ssubst)

  2234  apply (simp only: power_add)

  2235  apply simp_all

  2236 done

  2237

  2238 text {* by Brian Huffman *}

  2239 lemma zminus_zmod: "- ((x::int) mod m) mod m = - x mod m"

  2240 by (rule mod_minus_eq [symmetric])

  2241

  2242 lemma zdiff_zmod_left: "(x mod m - y) mod m = (x - y) mod (m::int)"

  2243 by (rule mod_diff_left_eq [symmetric])

  2244

  2245 lemma zdiff_zmod_right: "(x - y mod m) mod m = (x - y) mod (m::int)"

  2246 by (rule mod_diff_right_eq [symmetric])

  2247

  2248 lemmas zmod_simps =

  2249   mod_add_left_eq  [symmetric]

  2250   mod_add_right_eq [symmetric]

  2251   zmod_zmult1_eq   [symmetric]

  2252   mod_mult_left_eq [symmetric]

  2253   zpower_zmod

  2254   zminus_zmod zdiff_zmod_left zdiff_zmod_right

  2255

  2256 text {* Distributive laws for function @{text nat}. *}

  2257

  2258 lemma nat_div_distrib: "0 \<le> x \<Longrightarrow> nat (x div y) = nat x div nat y"

  2259 apply (rule linorder_cases [of y 0])

  2260 apply (simp add: div_nonneg_neg_le0)

  2261 apply simp

  2262 apply (simp add: nat_eq_iff pos_imp_zdiv_nonneg_iff zdiv_int)

  2263 done

  2264

  2265 (*Fails if y<0: the LHS collapses to (nat z) but the RHS doesn't*)

  2266 lemma nat_mod_distrib:

  2267   "\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> nat (x mod y) = nat x mod nat y"

  2268 apply (case_tac "y = 0", simp)

  2269 apply (simp add: nat_eq_iff zmod_int)

  2270 done

  2271

  2272 text  {* transfer setup *}

  2273

  2274 lemma transfer_nat_int_functions:

  2275     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) div (nat y) = nat (x div y)"

  2276     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) mod (nat y) = nat (x mod y)"

  2277   by (auto simp add: nat_div_distrib nat_mod_distrib)

  2278

  2279 lemma transfer_nat_int_function_closures:

  2280     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x div y >= 0"

  2281     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x mod y >= 0"

  2282   apply (cases "y = 0")

  2283   apply (auto simp add: pos_imp_zdiv_nonneg_iff)

  2284   apply (cases "y = 0")

  2285   apply auto

  2286 done

  2287

  2288 declare TransferMorphism_nat_int [transfer add return:

  2289   transfer_nat_int_functions

  2290   transfer_nat_int_function_closures

  2291 ]

  2292

  2293 lemma transfer_int_nat_functions:

  2294     "(int x) div (int y) = int (x div y)"

  2295     "(int x) mod (int y) = int (x mod y)"

  2296   by (auto simp add: zdiv_int zmod_int)

  2297

  2298 lemma transfer_int_nat_function_closures:

  2299     "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x div y)"

  2300     "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x mod y)"

  2301   by (simp_all only: is_nat_def transfer_nat_int_function_closures)

  2302

  2303 declare TransferMorphism_int_nat [transfer add return:

  2304   transfer_int_nat_functions

  2305   transfer_int_nat_function_closures

  2306 ]

  2307

  2308 text{*Suggested by Matthias Daum*}

  2309 lemma int_div_less_self: "\<lbrakk>0 < x; 1 < k\<rbrakk> \<Longrightarrow> x div k < (x::int)"

  2310 apply (subgoal_tac "nat x div nat k < nat x")

  2311  apply (simp (asm_lr) add: nat_div_distrib [symmetric])

  2312 apply (rule Divides.div_less_dividend, simp_all)

  2313 done

  2314

  2315 text {* code generator setup *}

  2316

  2317 context ring_1

  2318 begin

  2319

  2320 lemma of_int_num [code]:

  2321   "of_int k = (if k = 0 then 0 else if k < 0 then

  2322      - of_int (- k) else let

  2323        (l, m) = divmod_int k 2;

  2324        l' = of_int l

  2325      in if m = 0 then l' + l' else l' + l' + 1)"

  2326 proof -

  2327   have aux1: "k mod (2\<Colon>int) \<noteq> (0\<Colon>int) \<Longrightarrow>

  2328     of_int k = of_int (k div 2 * 2 + 1)"

  2329   proof -

  2330     have "k mod 2 < 2" by (auto intro: pos_mod_bound)

  2331     moreover have "0 \<le> k mod 2" by (auto intro: pos_mod_sign)

  2332     moreover assume "k mod 2 \<noteq> 0"

  2333     ultimately have "k mod 2 = 1" by arith

  2334     moreover have "of_int k = of_int (k div 2 * 2 + k mod 2)" by simp

  2335     ultimately show ?thesis by auto

  2336   qed

  2337   have aux2: "\<And>x. of_int 2 * x = x + x"

  2338   proof -

  2339     fix x

  2340     have int2: "(2::int) = 1 + 1" by arith

  2341     show "of_int 2 * x = x + x"

  2342     unfolding int2 of_int_add left_distrib by simp

  2343   qed

  2344   have aux3: "\<And>x. x * of_int 2 = x + x"

  2345   proof -

  2346     fix x

  2347     have int2: "(2::int) = 1 + 1" by arith

  2348     show "x * of_int 2 = x + x"

  2349     unfolding int2 of_int_add right_distrib by simp

  2350   qed

  2351   from aux1 show ?thesis by (auto simp add: divmod_int_mod_div Let_def aux2 aux3)

  2352 qed

  2353

  2354 end

  2355

  2356 lemma zmod_eq_dvd_iff: "(x::int) mod n = y mod n \<longleftrightarrow> n dvd x - y"

  2357 proof

  2358   assume H: "x mod n = y mod n"

  2359   hence "x mod n - y mod n = 0" by simp

  2360   hence "(x mod n - y mod n) mod n = 0" by simp

  2361   hence "(x - y) mod n = 0" by (simp add: mod_diff_eq[symmetric])

  2362   thus "n dvd x - y" by (simp add: dvd_eq_mod_eq_0)

  2363 next

  2364   assume H: "n dvd x - y"

  2365   then obtain k where k: "x-y = n*k" unfolding dvd_def by blast

  2366   hence "x = n*k + y" by simp

  2367   hence "x mod n = (n*k + y) mod n" by simp

  2368   thus "x mod n = y mod n" by (simp add: mod_add_left_eq)

  2369 qed

  2370

  2371 lemma nat_mod_eq_lemma: assumes xyn: "(x::nat) mod n = y  mod n" and xy:"y \<le> x"

  2372   shows "\<exists>q. x = y + n * q"

  2373 proof-

  2374   from xy have th: "int x - int y = int (x - y)" by simp

  2375   from xyn have "int x mod int n = int y mod int n"

  2376     by (simp add: zmod_int[symmetric])

  2377   hence "int n dvd int x - int y" by (simp only: zmod_eq_dvd_iff[symmetric])

  2378   hence "n dvd x - y" by (simp add: th zdvd_int)

  2379   then show ?thesis using xy unfolding dvd_def apply clarsimp apply (rule_tac x="k" in exI) by arith

  2380 qed

  2381

  2382 lemma nat_mod_eq_iff: "(x::nat) mod n = y mod n \<longleftrightarrow> (\<exists>q1 q2. x + n * q1 = y + n * q2)"

  2383   (is "?lhs = ?rhs")

  2384 proof

  2385   assume H: "x mod n = y mod n"

  2386   {assume xy: "x \<le> y"

  2387     from H have th: "y mod n = x mod n" by simp

  2388     from nat_mod_eq_lemma[OF th xy] have ?rhs

  2389       apply clarify  apply (rule_tac x="q" in exI) by (rule exI[where x="0"], simp)}

  2390   moreover

  2391   {assume xy: "y \<le> x"

  2392     from nat_mod_eq_lemma[OF H xy] have ?rhs

  2393       apply clarify  apply (rule_tac x="0" in exI) by (rule_tac x="q" in exI, simp)}

  2394   ultimately  show ?rhs using linear[of x y] by blast

  2395 next

  2396   assume ?rhs then obtain q1 q2 where q12: "x + n * q1 = y + n * q2" by blast

  2397   hence "(x + n * q1) mod n = (y + n * q2) mod n" by simp

  2398   thus  ?lhs by simp

  2399 qed

  2400

  2401 lemma div_nat_number_of [simp]:

  2402      "(number_of v :: nat)  div  number_of v' =

  2403           (if neg (number_of v :: int) then 0

  2404            else nat (number_of v div number_of v'))"

  2405   unfolding nat_number_of_def number_of_is_id neg_def

  2406   by (simp add: nat_div_distrib)

  2407

  2408 lemma one_div_nat_number_of [simp]:

  2409      "Suc 0 div number_of v' = nat (1 div number_of v')"

  2410 by (simp del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric])

  2411

  2412 lemma mod_nat_number_of [simp]:

  2413      "(number_of v :: nat)  mod  number_of v' =

  2414         (if neg (number_of v :: int) then 0

  2415          else if neg (number_of v' :: int) then number_of v

  2416          else nat (number_of v mod number_of v'))"

  2417   unfolding nat_number_of_def number_of_is_id neg_def

  2418   by (simp add: nat_mod_distrib)

  2419

  2420 lemma one_mod_nat_number_of [simp]:

  2421      "Suc 0 mod number_of v' =

  2422         (if neg (number_of v' :: int) then Suc 0

  2423          else nat (1 mod number_of v'))"

  2424 by (simp del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric])

  2425

  2426 lemmas dvd_eq_mod_eq_0_number_of =

  2427   dvd_eq_mod_eq_0 [of "number_of x" "number_of y", standard]

  2428

  2429 declare dvd_eq_mod_eq_0_number_of [simp]

  2430

  2431

  2432 subsubsection {* Code generation *}

  2433

  2434 definition pdivmod :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where

  2435   "pdivmod k l = (\<bar>k\<bar> div \<bar>l\<bar>, \<bar>k\<bar> mod \<bar>l\<bar>)"

  2436

  2437 lemma pdivmod_posDivAlg [code]:

  2438   "pdivmod k l = (if l = 0 then (0, \<bar>k\<bar>) else posDivAlg \<bar>k\<bar> \<bar>l\<bar>)"

  2439 by (subst posDivAlg_div_mod) (simp_all add: pdivmod_def)

  2440

  2441 lemma divmod_int_pdivmod: "divmod_int k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else

  2442   apsnd ((op *) (sgn l)) (if 0 < l \<and> 0 \<le> k \<or> l < 0 \<and> k < 0

  2443     then pdivmod k l

  2444     else (let (r, s) = pdivmod k l in

  2445       if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"

  2446 proof -

  2447   have aux: "\<And>q::int. - k = l * q \<longleftrightarrow> k = l * - q" by auto

  2448   show ?thesis

  2449     by (simp add: divmod_int_mod_div pdivmod_def)

  2450       (auto simp add: aux not_less not_le zdiv_zminus1_eq_if

  2451       zmod_zminus1_eq_if zdiv_zminus2_eq_if zmod_zminus2_eq_if)

  2452 qed

  2453

  2454 lemma divmod_int_code [code]: "divmod_int k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else

  2455   apsnd ((op *) (sgn l)) (if sgn k = sgn l

  2456     then pdivmod k l

  2457     else (let (r, s) = pdivmod k l in

  2458       if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"

  2459 proof -

  2460   have "k \<noteq> 0 \<Longrightarrow> l \<noteq> 0 \<Longrightarrow> 0 < l \<and> 0 \<le> k \<or> l < 0 \<and> k < 0 \<longleftrightarrow> sgn k = sgn l"

  2461     by (auto simp add: not_less sgn_if)

  2462   then show ?thesis by (simp add: divmod_int_pdivmod)

  2463 qed

  2464

  2465 code_modulename SML

  2466   Divides Arith

  2467

  2468 code_modulename OCaml

  2469   Divides Arith

  2470

  2471 code_modulename Haskell

  2472   Divides Arith

  2473

  2474 end
`